:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> m = \\frac{9300 \\, \\text{g}}{1000} = 9.3 \\, \\text{kg} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Calculate the force due to gravity:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> F = m \\times g = 9.3 \\, \\text{kg} \\times 9.81 \\, \\text{m/s}^2 = 91.233 \\, \\text{N} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Convert the length of the hammer from centimeters to meters:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 18 \\, \\text{cm} = 0.18 \\, \\text{m} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Calculate the torque:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\tau = 0.18 \\, \\text{m} \\times 91.233 \\, \\text{N} = 16.42194 \\, \\text{Nm} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nThus, the torque about the pivot point due to the weight of the screw at the end of the hammer handle is approximately \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 16.42 \\, \\text{Nm} \u003C/math-field>\u003C/math-field>.",484,97,"suppose-you-have-to-use-a-hammer-in-your-hand-on-a-wall-with-a-screw-of-mass-9300-g-the-hammer-is-18-cm-long-calculate-the-torque-of-the-screw",{"id":51,"category":36,"text_question":52,"photo_question":38,"text_answer":53,"step_text_answer":8,"step_photo_answer":8,"views":54,"likes":55,"slug":56},538046," A cylinder makes six turns in 2 seconds, calculate: a) its angular velocity in rad/s; b) its period and c) its frequency.","a) Angular velocity $\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\omega\u003C/math-field>\u003C/math-field>$ is given by the formula:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\omega = \\frac{\\Delta \\theta}{\\Delta t}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nHere, the cylinder makes 6 turns, and each turn is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2\\pi\u003C/math-field>\u003C/math-field> radians. Therefore, in 6 turns, the angle in radians is:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\Delta \\theta = 6 \\times 2\\pi = 12\\pi \\text{ radians}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nThe time period $\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\Delta t\u003C/math-field>\u003C/math-field>$ is 2 seconds, so the angular velocity is:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\omega = \\frac{12\\pi}{2} = 6\\pi \\text{ rad/s}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nb) The period $\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>T\u003C/math-field>\u003C/math-field>$ is the time it takes to complete one full rotation $1 turn$. Since the cylinder makes 6 turns in 2 seconds:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>T = \\frac{\\Delta t}{\\text{number of turns}} = \\frac{2}{6} = \\frac{1}{3} \\text{ s}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nc) The frequency $\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>f\u003C/math-field>\u003C/math-field>$ is the reciprocal of the period:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>f = \\frac{1}{T} = \\frac{1}{\\frac{1}{3}} = 3 \\text{ Hz}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nThus, the answers are:\u003Cbr />\n\u003Cbr />\na) \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\omega = 6\\pi \\text{ rad/s}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nb) \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>T = \\frac{1}{3} \\text{ s}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nc) \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>f = 3 \\text{ Hz}\u003C/math-field>\u003C/math-field>",250,50,"a-cylinder-makes-six-turns-in-2-seconds-calculate-a-its-angular-velocity-in-rad-s-b-its-period-and-c-its-frequency",{"id":58,"category":36,"text_question":59,"photo_question":38,"text_answer":60,"step_text_answer":8,"step_photo_answer":8,"views":61,"likes":62,"slug":63},538045," A 73,000 g Ferris wheel accelerates from rest to an angular speed of 6.2 rad/s in 2 minutes. Considering the wheel as a hollow circular disk of radius 200 cm, calculate the net force on it?","1. Convert mass from grams to kilograms: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> m = 73,000 \\, \\text{g} = 73 \\, \\text{kg} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Convert radius from centimeters to meters:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> r = 200 \\, \\text{cm} = 2 \\, \\text{m} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Convert time from minutes to seconds: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> t = 2 \\, \\text{minutes} = 120 \\, \\text{seconds} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Calculate angular acceleration:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\alpha = \\frac{\\omega_f - \\omega_i}{t} = \\frac{6.2 \\, \\text{rad/s} - 0 \\, \\text{rad/s}}{120 \\, \\text{s}} = 0.0517 \\, \\text{rad/s}^2 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>5. Moment of inertia of a hollow circular disk:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> I = m \\cdot r^2 = 73 \\, \\text{kg} \\cdot $2 \\, \\text{m}$^2 = 292 \\, \\text{kg} \\cdot \\text{m}^2 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>6. Calculate net torque:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\tau = I \\cdot \\alpha = 292 \\, \\text{kg} \\cdot \\text{m}^2 \\times 0.0517 \\, \\text{rad/s}^2 = 15.1044 \\, \\text{N} \\cdot \\text{m} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>7. Calculate net force $since torque = force × radius$:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> F = \\frac{\\tau}{r} = \\frac{15.1044 \\, \\text{N} \\cdot \\text{m}}{2 \\, \\text{m}} = 7.5522 \\, \\text{N} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Rounding to a sensible number of significant figures gives the net force:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>F\\approx7.55\\,\\text{N}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Therefore, the net force on the Ferris wheel is approximately \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>7.55\\,\\text{N}\u003C/math-field>\u003C/math-field> .",655,131,"a-73-000-g-ferris-wheel-accelerates-from-rest-to-an-angular-speed-of-6-2-rad-s-in-2-minutes-considering-the-wheel-as-a-hollow-circular-disk-of-radius-200-cm-calculate-the-net-force-on-it",{"id":65,"category":36,"text_question":66,"photo_question":38,"text_answer":67,"step_text_answer":8,"step_photo_answer":8,"views":68,"likes":69,"slug":70},538044," A cylindrical piece of material 12 in in diameter is rotated on a lathe at 1300 rev/min. What is the tangential velocity at the surface of the cylinder? Value 4 points","1. Find angular velocity: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\omega = 1300 \\times 2\\pi \\text{ radians per minute} \u003C/math-field>\u003C/math-field>\u003Cbr />\n2. Calculate the radius: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> r = \\frac{12}{2} = 6 \\text{ inches} \u003C/math-field>\u003C/math-field>\u003Cbr />\n3. Use the formula for tangential velocity:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v = \\omega \\cdot r = $1300 \\times 2\\pi$ \\cdot 6 = 15600\\pi \\text{ inches per minute} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nAnswer: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v = 15600\\pi \\text{ inches per minute} \u003C/math-field>\u003C/math-field>",649,130,"a-cylindrical-piece-of-material-12-in-in-diameter-is-rotated-on-a-lathe-at-1300-rev-min-what-is-the-tangential-velocity-at-the-surface-of-the-cylinder-value-4-points",{"id":72,"category":36,"text_question":73,"photo_question":38,"text_answer":74,"step_text_answer":8,"step_photo_answer":8,"views":75,"likes":76,"slug":77},538043," A pulley in a well with a diameter of 1200 cm initially rotates at 1.3 rev/s and then receives a constant angular acceleration of 3.12 rad/s2 . What is the tangential velocity of a belt mounted on said pulley? After 1 second, what is the tangential acceleration of the belt?","1. Calculate the initial angular velocity in radians per second: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\omega_0 = 1.3 \\, \\text{rev/s} \\times 2\\pi \\, \\text{rad/rev} = 2.6\\pi \\, \\text{rad/s} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Determine the final angular velocity after 1 second using the equation for angular velocity with constant acceleration:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\omega = \\omega_0 + \\alpha t \u003C/math-field>\u003C/math-field>\u003Cbr />\n where \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\alpha = 3.12 \\, \\text{rad/s}^2 \u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> t = 1 \\, \\text{s} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Calculate \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\omega \u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\omega = 2.6\\pi + 3.12 \\times 1 = 2.6\\pi + 3.12 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n4. Calculate the radius of the pulley:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> r = \\frac{1200 \\, \\text{cm}}{2} = 600 \\, \\text{cm} = 6 \\, \\text{m} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Find the tangential velocity \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v \u003C/math-field>\u003C/math-field> at \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> t = 1 \\, \\text{s} \u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v = \\omega \\times r \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v = $2.6\\pi + 3.12$ \\times 6 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Simplify to find \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v \u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v = $8.168 + 3.12$ \\times 6 = 11.288 \\times 6 = 67.728 \\, \\text{m/s} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n7. Calculate the tangential acceleration \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> a_t \u003C/math-field>\u003C/math-field>, which is constant:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> a_t = \\alpha \\times r = 3.12 \\times 6 = 18.72 \\, \\text{m/s}^2 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n8. Final answers: \u003Cbr />\n Tangential velocity after 1 second: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 67.728 \\, \\text{m/s} \u003C/math-field>\u003C/math-field>\u003Cbr />\n Tangential acceleration: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 18.72 \\, \\text{m/s}^2 \u003C/math-field>\u003C/math-field>",955,191,"a-pulley-in-a-well-with-a-diameter-of-1200-cm-initially-rotates-at-1-3-rev-s-and-then-receives-a-constant-angular-acceleration-of-3-12-rad-s2-what-is-the-tangential-velocity-of-a-belt-mounted-on-s",{"id":79,"category":36,"text_question":80,"photo_question":38,"text_answer":81,"step_text_answer":8,"step_photo_answer":8,"views":82,"likes":83,"slug":84},538041,"Why do we use multiplication when dividing fractions. I know how to do the steps, but why do you keep the first number, changed to multiplication, and then write the reciprocal of the fraction.\n\nI guess I just don’t understand why we switched to multiplication. Thank you.","1. **Keep the First Fraction:** Write the first fraction as it is. \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{a}{b}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. **Change the Operation:** Change the division sign to a multiplication sign. \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\times\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. **Use the Reciprocal:** Use the reciprocal of the second fraction $flip the numerator and denominator$. \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{d}{c}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. **Multiply the Fractions:** Multiply the numerators and denominators. \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{a \\times d}{b \\times c}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. **Answer:** The division of the two given fractions as multiplication results in: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{a \\times d}{b \\times c}\u003C/math-field>\u003C/math-field>",702,140,"why-do-we-use-multiplication-when-dividing-fractions-i-know-how-to-do-the-steps-but-why-do-you-keep-the-first-number-changed-to-multiplication-and-then-write-the-reciprocal-of-the-fraction-i-gu",{"id":86,"category":36,"text_question":87,"photo_question":38,"text_answer":88,"step_text_answer":8,"step_photo_answer":8,"views":89,"likes":90,"slug":91},538040,"Prove the trig identity:\n\nSec^2x-Sin^2xSec^2x=1","1. Start with the left-hand side of the equation: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sec^2{x} - \\sin^2{x} \\sec^2{x} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Factor out \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sec^2{x} \u003C/math-field>\u003C/math-field>: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sec^2{x} $1 - \\sin^2{x}$ \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Use the Pythagorean identity \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sin^2{x} + \\cos^2{x} = 1 \u003C/math-field>\u003C/math-field> to replace \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 1 - \\sin^2{x} \u003C/math-field>\u003C/math-field> with \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\cos^2{x} \u003C/math-field>\u003C/math-field>: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sec^2{x} \\cdot \\cos^2{x} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Substitute \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sec{x} = \\frac{1}{\\cos{x}} \u003C/math-field>\u003C/math-field>: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\left$\\frac{1}{\\cos^2{x}}\\right$ \\cdot \\cos^2{x} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Simplify: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 1 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nTherefore, the identity is proven: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sec^2{x} - \\sin^2{x} \\sec^2{x} = 1 \u003C/math-field>\u003C/math-field>",294,59,"prove-the-trig-identity-sec-2x-sin-2xsec-2x-1",{"id":93,"category":36,"text_question":94,"photo_question":38,"text_answer":95,"step_text_answer":8,"step_photo_answer":8,"views":96,"likes":97,"slug":98},538039,"Prove the trig identity:\n\n1+Sec^2x/Sec^2x = 1 + cos^2x","1. Start with the left-hand side of the equation: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1+\\sec^2 x}{\\sec^2 x}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Recall that \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sec x = \\frac{1}{\\cos x}\u003C/math-field>\u003C/math-field> , so \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sec^2 x = \\frac{1}{\\cos^2 x}\u003C/math-field>\u003C/math-field> . Therefore, substitute:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1+\\frac{1}{\\cos^2 x}}{\\frac{1}{\\cos^2 x}}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Simplify the fraction:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{\\cos^2x+1}{\\frac{\\cos^2\\left$x\\right$}{\\cos^2x}}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Multiply by the reciprocal of the denominator:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$\\cos^2x+1$\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>6. Simplify to get:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\cos^2 x + 1\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>7. Compare with the right-hand side, which is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1 + \\cos^2 x\u003C/math-field>\u003C/math-field> . Since \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\cos^2 x + 1\u003C/math-field>\u003C/math-field> is equivalent to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1 + \\cos^2 x\u003C/math-field>\u003C/math-field> , the trigonometric identity is proved.\u003Cbr>\u003Cbr>Answer: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1 + \\cos^2 x\u003C/math-field>\u003C/math-field>",1031,206,"prove-the-trig-identity-1-sec-2x-sec-2x-1-cos-2x",{"id":100,"category":36,"text_question":101,"photo_question":38,"text_answer":102,"step_text_answer":8,"step_photo_answer":8,"views":103,"likes":104,"slug":105},538038,"Prove the trig identity:\n\nCos x/1-sinx - Cos x/1+sin x = 2tan x","\u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$=\\frac{\\sin(2x)}{(-\\sin(x)+1)(\\sin(x)+1)}$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$=\\frac{\\sin(2x)}{\\cos^{2}(x)}$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$=2\\tan(x)$\u003C/math-field>\n \u003Cbr>\n \u003C/div>",770,154,"prove-the-trig-identity-cos-x-1-sinx-cos-x-1-sin-x-2tan-x",{"id":107,"category":36,"text_question":108,"photo_question":38,"text_answer":109,"step_text_answer":8,"step_photo_answer":8,"views":110,"likes":111,"slug":112},538037,"Prove the trig identity:\n\n1+cos x/sin x = csc x + cot x","1. Start with the right-hand side of the equation: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\csc x + \\cot x \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cbr />\n2. Express \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\csc x \u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\cot x \u003C/math-field>\u003C/math-field> in terms of sine and cosine:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\csc x = \\frac{1}{\\sin x} \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\cot x = \\frac{\\cos x}{\\sin x} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Add the fractions:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\csc x + \\cot x = \\frac{1}{\\sin x} + \\frac{\\cos x}{\\sin x} = \\frac{1 + \\cos x}{\\sin x} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. The expression \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{1 + \\cos x}{\\sin x} \u003C/math-field>\u003C/math-field> matches the left-hand side of the identity.\u003Cbr />\n\u003Cbr />\nTherefore, the identity is valid: \u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{1 + \\cos x}{\\sin x} = \\csc x + \\cot x \u003C/math-field>\u003C/math-field>",605,121,"prove-the-trig-identity-1-cos-x-sin-x-csc-x-cot-x",{"id":114,"category":36,"text_question":115,"photo_question":38,"text_answer":116,"step_text_answer":8,"step_photo_answer":8,"views":117,"likes":118,"slug":119},538036,"Prove the trig identity:\n\n $1-cos θ$$1+cos θ$= 1/csc^2 θ","1. Start with the left side of the equation: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$1 - \\cos \\theta$$1 + \\cos \\theta$\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Apply the difference of squares formula: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$1 - \\cos \\theta$$1 + \\cos \\theta$ = 1 - \\cos^2 \\theta\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Use the Pythagorean identity: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1 - \\cos^2 \\theta = \\sin^2 \\theta\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Relate to the right side: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sin^2 \\theta = \\frac{1}{\\csc^2 \\theta}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>",1096,219,"prove-the-trig-identity-1-cos-1-cos-1-csc-2",{"id":121,"category":36,"text_question":122,"photo_question":38,"text_answer":123,"step_text_answer":8,"step_photo_answer":8,"views":124,"likes":125,"slug":126},538035,"Prove the trig identity:\n\n Sec θ - Cos θ/Sec θ = Sin^2 θ","1. Start with the left-hand side $LHS$: \u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{LHS} = \\frac{\\sec \\theta - \\cos \\theta}{\\sec \\theta}\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n2. Rewrite the terms in terms of sine and cosine:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sec \\theta = \\frac{1}{\\cos \\theta}\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n3. Substitute into the LHS:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{LHS} = \\frac{\\frac{1}{\\cos \\theta} - \\cos \\theta}{\\frac{1}{\\cos \\theta}}\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n4. Simplify the expression inside the fraction:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= \\frac{\\frac{1 - \\cos^2 \\theta}{\\cos \\theta}}{\\frac{1}{\\cos \\theta}}\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n5. Simplify further by multiplying by the reciprocal:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= $1 - \\cos^2 \\theta$\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Substitute the Pythagorean identity:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= \\sin^2 \\theta\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\nHence, the identity is proven since:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{\\sec \\theta - \\cos \\theta}{\\sec \\theta} = \\sin^2 \\theta\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\nSo, the right-hand side $RHS$ and transformed LHS are equal. \u003Cbr />\n\u003Cbr />\nThis completes the proof of the identity.",1434,287,"prove-the-trig-identity-sec-cos-sec-sin-2",{"id":128,"category":36,"text_question":129,"photo_question":38,"text_answer":130,"step_text_answer":8,"step_photo_answer":8,"views":131,"likes":132,"slug":133},538032,"How much is 18% of a kilometer?","18% of a kilometer can be calculated by multiplying 18% with the length of a kilometer. \u003Cbr />\n\u003Cbr />\n[SOLUTION] \u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 180 \\text{ meters} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n[STEP-BY-STEP]\u003Cbr />\n\u003Cbr />\n1. Convert percentage to a decimal.\u003Cbr />\n - 18% is the same as 0.18.\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 18\\% = \\frac{18}{100} = 0.18 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. A kilometer is 1,000 meters.\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 1 \\text{ kilometer} = 1000 \\text{ meters} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Calculate 18% of a kilometer in meters.\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 0.18 \\times 1000 = 180 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Therefore, 18% of a kilometer is:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 180 \\text{ meters} \u003C/math-field>\u003C/math-field>",520,104,"how-much-is-18-of-a-kilometer",{"id":135,"category":36,"text_question":136,"photo_question":38,"text_answer":137,"step_text_answer":8,"step_photo_answer":8,"views":138,"likes":139,"slug":140},538031,"How much is 10% of a kilogram?","1. A kilogram is equal to 1000 grams. \u003Cbr />\n\u003Cbr />\n2. To find 10% of a kilogram, calculate 10% of 1000 grams: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.1 \\times 1000 = 100\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n3. Convert the result back to kilograms:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>100 \\, \\text{grams} = 0.1 \\, \\text{kg}\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\nThus, 10% of a kilogram is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.1 \\, \\text{kg}\u003C/math-field>\u003C/math-field>.",763,153,"how-much-is-10-of-a-kilogram",{"id":142,"category":36,"text_question":143,"photo_question":38,"text_answer":144,"step_text_answer":8,"step_photo_answer":8,"views":145,"likes":146,"slug":147},538030,"98/100 \nWrite the fraction as a decimal","1. Begin with the fraction \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{98}{100}\u003C/math-field>\u003C/math-field>. \u003Cbr />\n\u003Cbr />\n2. Convert the fraction to a decimal by dividing the numerator by the denominator: \u003Cbr />\n \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{98}{100} = 98 \\div 100 = 0.98\u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cbr />\n3. Thus, the fraction \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{98}{100}\u003C/math-field>\u003C/math-field> is equivalent to the decimal 0.98.\u003Cbr />\n\u003Cbr />\nAnswer: 0.98",383,77,"98-100-write-the-fraction-as-a-decimal",{"id":149,"category":36,"text_question":150,"photo_question":38,"text_answer":151,"step_text_answer":8,"step_photo_answer":8,"views":152,"likes":153,"slug":154},538029,"$-3,-2$ and $-8, -2$\nFind the distance between 2 points","To find the distance between two points \\((-3,-2)\\) and \\((-8, -2)\\) in a 2D coordinate plane, we use the distance formula:\u003Cbr>\u003Cbr>1. The distance formula is: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Substitute the given points \\((-3, -2)\\) and \\((-8, -2)\\) into the formula:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> d = \\sqrt{((-8) - (-3))^2 + ((-2) - (-2))^2} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Simplify the expression inside the square root:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> d = \\sqrt{(-8 + 3)^2 + (0)^2} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> d = \\sqrt{(-5)^2 + 0} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Calculate:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> d = \\sqrt{25} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>5. Hence, the distance between the points is:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5\u003C/math-field>\u003C/math-field> units",1078,216,"3-2-and-8-2-find-the-distance-between-2-points",{"id":156,"category":36,"text_question":157,"photo_question":38,"text_answer":158,"step_text_answer":8,"step_photo_answer":8,"views":159,"likes":160,"slug":161},538028,"2.01c-6=-0.15c+6.96","Solution:\u003Cbr />\n1. Start with the given equation:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2.01c - 6 = -0.15c + 6.96\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Move all terms involving \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c\u003C/math-field>\u003C/math-field> to one side by adding \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.15c\u003C/math-field>\u003C/math-field> to both sides:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2.01c + 0.15c = 6.96 + 6\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Simplify the equation:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2.16c = 12.96\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c\u003C/math-field>\u003C/math-field> by dividing both sides by \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2.16\u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c = \\frac{12.96}{2.16}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Calculate the value:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c = 6\u003C/math-field>\u003C/math-field>",865,173,"2-01c-6-0-15c-6-96",{"id":163,"category":36,"text_question":164,"photo_question":38,"text_answer":165,"step_text_answer":8,"step_photo_answer":8,"views":166,"likes":167,"slug":168},538027,"1/4x+13>0.25(2x-32)","Solution:\u003Cbr />\n1. Given inequality:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{4}x + 13 > 0.25(2x - 32)\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Simplify the right side:\u003Cbr />\n * Convert 0.25 to a fraction: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.25 = \\frac{1}{4}\u003C/math-field>\u003C/math-field>\u003Cbr />\n * Distribute: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{4}(2x - 32) = \\frac{1}{4} \\cdot 2x - \\frac{1}{4} \\cdot 32 = \\frac{1}{2}x - 8\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Rewrite the inequality:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{4}x + 13 > \\frac{1}{2}x - 8\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Eliminate fractions by multiplying all terms by 4:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x + 52 > 2x - 32\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Rearrange terms:\u003Cbr />\n * Subtract \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> from both sides:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>52 > x - 32\u003C/math-field>\u003C/math-field>\u003Cbr />\n * Add 32 to both sides:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>84 > x\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. The solution to the inequality is:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x \u003C 84\u003C/math-field>\u003C/math-field>",301,60,"1-4x-13-0-25-2x-32",{"id":170,"category":36,"text_question":171,"photo_question":38,"text_answer":172,"step_text_answer":8,"step_photo_answer":8,"views":173,"likes":174,"slug":175},538026,"0.5(4x+24)=22x-2","Solution:\u003Cbr />\n1. Given equation:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.5(4x + 24) = 22x - 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Simplify the left side:\u003Cbr />\n- Apply the distributive property: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.5 \\times 4x + 0.5 \\times 24 = 22x - 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n- This gives us: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2x + 12 = 22x - 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Rearrange to solve for x:\u003Cbr />\n- Subtract \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2x\u003C/math-field>\u003C/math-field> from both sides: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2x + 12 - 2x = 22x - 2 - 2x\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Simplify: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>12 = 20x - 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Add 2 to both sides to isolate terms:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>12 + 2 = 20x - 2 + 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Simplify: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>14 = 20x\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Divide by 20 to solve for x:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{14}{20}\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Simplify the fraction: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{7}{10}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nThe solution to the equation is $x = \\\\frac{7}{10}$.",614,123,"0-5-4x-24-22x-2",{"first":6,"last":177,"prev":8,"next":10},186,{"current_page":6,"from":6,"last_page":177,"links":179,"path":213,"per_page":214,"to":214,"total":215},[180,183,186,188,190,192,194,197,200,203,206,209,211],{"url":6,"label":181,"active":182},"1",true,{"url":10,"label":184,"active":185},"2",false,{"url":13,"label":187,"active":185},"3",{"url":16,"label":189,"active":185},"4",{"url":19,"label":191,"active":185},"5",{"url":22,"label":193,"active":185},"6",{"url":195,"label":196,"active":185},7,"7",{"url":198,"label":199,"active":185},8,"8",{"url":201,"label":202,"active":185},9,"9",{"url":204,"label":205,"active":185},10,"10",{"url":207,"label":208,"active":185},185,"185",{"url":177,"label":210,"active":185},"186",{"url":10,"label":212,"active":185},"Next »","https://api.math-master.org/api/question",20,3704,{"data":217},{"questions":218},[219,223,227,231,235,239,243,247,251,255,259,263,267,271,275,279,283,287,291,295],{"id":220,"category":36,"text_question":221,"slug":222},532064,"12-6x=4x+2","12-6x-4x-2",{"id":224,"category":36,"text_question":225,"slug":226},532090,"String x = 5\r\nInt y=2\r\nSystem.out.println(x+y)","string-x-5-int-y-2-system-out-println-x-y",{"id":228,"category":36,"text_question":229,"slug":230},533892,"I) Find the directional derivative of 𝑓(𝑥, 𝑦) = 𝑥 sin 𝑦 at (1,0) in the direction of the unit vector that make an angle of 𝜋/4 with positive 𝑥-axis.","i-find-the-directional-derivative-of-f-x-y-x-sin-y-at-1-0-in-the-direction-of-the-unit-vector-that-make-an-angle-of-4-with-positive-x-axis",{"id":232,"category":36,"text_question":233,"slug":234},533906,"If L = (-2, -5) is reflected across y= -4 , what are the coordinates of L?","if-l-2-5-is-reflected-across-y-4-what-are-the-coordinates-of-l",{"id":236,"category":36,"text_question":237,"slug":238},533915,"3x+5y=11\r\n2x-3y=1","3x-5y-11-2x-3y-1",{"id":240,"category":36,"text_question":241,"slug":242},533948,"Derivative of x squared","derivative-of-x-squared",{"id":244,"category":36,"text_question":245,"slug":246},534055,"7/6-(-1/9)","7-6-1-9",{"id":248,"category":36,"text_question":249,"slug":250},534081,"Mrs. Emily saved RM10000 in a bank. At the end of the eighth year, the amount of money accumulated amounted to RM19992.71. If the bank pays an annual interest of x% for a year compounded every 6 months. Calculate the value of x.","mrs-emily-saved-rm10000-in-a-bank-at-the-end-of-the-eighth-year-the-amount-of-money-accumulated-amounted-to-rm19992-71-if-the-bank-pays-an-annual-interest-of-x-for-a-year-compounded-every-6-month",{"id":252,"category":36,"text_question":253,"slug":254},534102,"logy/logx + logz/logy + logt/logz = 8x².t \r\nx=?","logy-logx-logz-logy-logt-logz-8x-t-x",{"id":256,"category":36,"text_question":257,"slug":258},534152,"Determine the reduced equation of the straight line that is perpendicular to the straight line r: y=4x-10 and passes through the origin of the Cartesian plane","determine-the-reduced-equation-of-the-straight-line-that-is-perpendicular-to-the-straight-line-r-y-4x-10-and-passes-through-the-origin-of-the-cartesian-plane",{"id":260,"category":36,"text_question":261,"slug":262},534208,"7. Find the equation of the line passing through the points (−4,−2) 𝑎𝑛𝑑 (3,6),\n\ngive the equation in the form 𝑎𝑥+𝑏𝑦+𝑐=0,\n\nwhere 𝑎,𝑏,𝑐 are whole numbers and 𝑎>0.","7-find-the-equation-of-the-line-passing-through-the-points-4-2-and-3-6-give-the-equation-in-the-form-ax-by-c-0-where-a-b-c-are-whole-numbers-and-a-0",{"id":264,"category":36,"text_question":265,"slug":266},534211,"The physician orders 15mg of tramadol(liquid). On hand is 30mg/2mL vials. How many mL will the MA administer?","the-physician-orders-15mg-of-tramadol-liquid-on-hand-is-30mg-2ml-vials-how-many-ml-will-the-ma-administer",{"id":268,"category":36,"text_question":269,"slug":270},534224,"Is -11/8 greater than or less than -1.37?","is-11-8-greater-than-or-less-than-1-37",{"id":272,"category":36,"text_question":273,"slug":274},534241,"Let A, B, C and D be sets such that | A| = |C| and |B| = |D|. Prove that |A × B| = |C × D|","let-a-b-c-and-d-be-sets-such-that-a-c-and-b-d-prove-that-a-b-c-d",{"id":276,"category":36,"text_question":277,"slug":278},534314,"Two minus log 3X equals log (X over 12)","two-minus-log-3x-equals-log-x-over-12",{"id":280,"category":36,"text_question":281,"slug":282},534483,"A property sold for $745,000 in a co-brokered transaction. The seller has agreed to pay a 7% commission to the listing firm. The listing firm has agreed to equally split the commission with the selling firm. If the buyer’s broker will receive 8% of the selling firm’s commission, how much commission will the buyer’s broker receive? \n$14,900\n$3725\n$$37250\n$18625","a-property-sold-for-745-000-in-a-co-brokered-transaction-the-seller-has-agreed-to-pay-a-7-commission-to-the-listing-firm-the-listing-firm-has-agreed-to-equally-split-the-commission-with-the-sellin",{"id":284,"category":36,"text_question":285,"slug":286},534503,"A factory produces glass for windows. The thickness X of an arbitrarily selected pane of glass is assumed to be \r\nNormally distributed with expectation μ = 4.10 and standard deviation σ = 0.04. Expectation and \r\nStandard deviation is measured in millimeters. \r\nWhat is the probability that an arbitrary route has a thickness less than 4.00 mm?","a-factory-produces-glass-for-windows-the-thickness-x-of-an-arbitrarily-selected-pane-of-glass-is-assumed-to-be-normally-distributed-with-expectation-u-4-10-and-standard-deviation-0-04-expect",{"id":288,"category":36,"text_question":289,"slug":290},534508,"Let G be the center of gravity of triangle ABC. We draw through A a parallel to BC on which we take a point D so that DG⊥BG. If the area of the quadrilateral AGBD is equal to s, show that AC·BD≥2·s.","let-g-be-the-center-of-gravity-of-triangle-abc-we-draw-through-a-a-parallel-to-bc-on-which-we-take-a-point-d-so-that-dg-bg-if-the-area-of-the-quadrilateral-agbd-is-equal-to-s-show-that-ac-bd-2-s",{"id":292,"category":36,"text_question":293,"slug":294},534516,"Write the inequality in the form of a\u003Cx\u003Cb.\n|x| \u003C c^2","write-the-inequality-in-the-form-of-a-x-b-x-c-2",{"id":296,"category":36,"text_question":297,"slug":298},534571,"A group of 17 people spent 9 days on vacation and spent R$776.34 on barbecue meat and the bill needs to be divided as follows: 6 people stayed for 9 days, 7 people stayed for 4 days, and 2 people stayed for 5 days and 2 people stayed 3 days, how much does each group have to pay for the days they stayed?","a-group-of-17-people-spent-9-days-on-vacation-and-spent-r-776-34-on-barbecue-meat-and-the-bill-needs-to-be-divided-as-follows-6-people-stayed-for-9-days-7-people-stayed-for-4-days-and-2-people-stay",{"data":300},{"id":301,"category":36,"slug":302,"text_question":303,"photo_question":8,"text_answer":304,"step_text_answer":8,"step_photo_answer":8,"views":305,"likes":306,"expert":307},533970,"two-numbers-differ-by-7-and-the-sum-of-their-squares-is-29-find-the-numbers","Two numbers differ by 7, and the sum of their squares is 29. Find the numbers.","let numbers are x and y.\nthen x-y=7...............(1) and \u003Cmath-field read-only> x^2+y^2=29 \u003C/math-field> \nFrom (1), \u003Cmath-field read-only> \\left(x-y\\right)^2=49 \u003C/math-field> \u003Cmath-field read-only> x^2+y^2-2xy=49 \u003C/math-field>\n 29-2xy=49\nxy=-10 then y=-10/x,\nNow from (1),\n\u003Cmath-field read-only> x+\\frac{10}{x}=7 \u003C/math-field>\u003Cmath-field read-only> x^2-7x+10=0 \u003C/math-field> so x=2,5, then y=-5,-2\nSo the numbers are 2,-5 or 5,-2",1400,280,{"id":19,"name":308,"photo":309,"biography":310,"created_at":8,"updated_at":8,"rating":311,"total_answer":132},"Rasheed","https://api.math-master.org/img/experts/5/5.webp","Born with an insatiable curiosity, I began my journey with mathematics during my school years. From solving basic arithmetic problems to unraveling complex algebraic equations, I developed a deep passion for the subject. Upon entering university, my learnings diving into advanced topics like abstract algebra, numerical analysis and differential equations.\nIn retrospect, my journey through the realm of mathematics has been an exhilarating adventure of discovery, learning, and growth.\n",4.7,{"data":313},{"questions":314},[315,319,323,327,331,335,339,343,347,351,355,359,363,367,371,375,379,383,387,391],{"id":316,"category":36,"text_question":317,"slug":318},532001,"A normal random variable x has a mean of 50 and a standard deviation of 10. Would it be unusual to see the value x = 0? Explain your answer.","a-normal-random-variable-x-has-a-mean-of-50-and-a-standard-deviation-of-10-would-it-be-unusual-to-see-the-value-x-0-explain-your-answer",{"id":320,"category":36,"text_question":321,"slug":322},532004,"Find 2 numbers that the sum of 1/3 of the first plus 1/5 of the second will be equal to 13 and that if you multiply the first by 5 and the second by 7 you get 247 as the sum of the two products\nwith replacement solution","find-2-numbers-that-the-sum-of-1-3-of-the-first-plus-1-5-of-the-second-will-be-equal-to-13-and-that-if-you-multiply-the-first-by-5-and-the-second-by-7-you-get-247-as-the-sum-of-the-two-products-with-r",{"id":324,"category":36,"text_question":325,"slug":326},532033,"𝑦 = ( 𝑥2 − 3) (𝑥3 + 2 𝑥 + 1)","y-x2-3-x3-2-x-1",{"id":328,"category":36,"text_question":329,"slug":330},532056,"5 squirrels were found to have an average weight of 9.3 ounces with a sample standard deviation is 1.1. Find the 95% confidence interval of the true mean weight","5-squirrels-were-found-to-have-an-average-weight-of-9-3-ounces-with-a-sample-standard-deviation-is-1-1-find-the-95-confidence-interval-of-the-true-mean-weight",{"id":332,"category":36,"text_question":333,"slug":334},533908,"Evaluate limx→∞tan−1(x)\r\n using that y=tan−1(x)\r\n exactly when x=tan(y)\r\n. (Hint: Both tan\r\n and tan−1\r\n are continuous!)","evaluate-limx-tan-1-x-using-that-y-tan-1-x-exactly-when-x-tan-y-hint-both-tan-and-tan-1-are-continuous",{"id":336,"category":36,"text_question":337,"slug":338},533971,"The sum of two numbers is 6, and the sum of their squares is 28. Find these numbers exactly","the-sum-of-two-numbers-is-6-and-the-sum-of-their-squares-is-28-find-these-numbers-exactly",{"id":340,"category":36,"text_question":341,"slug":342},534007,"A soft drink machine outputs a mean of 23 ounces per cup. The machines output is normally distributed with a standard deviation of 3 ounces. What is the probability of filling a cup between 26 and 28 ounces round your answer to four decimal places","a-soft-drink-machine-outputs-a-mean-of-23-ounces-per-cup-the-machines-output-is-normally-distributed-with-a-standard-deviation-of-3-ounces-what-is-the-probability-of-filling-a-cup-between-26-and-28",{"id":344,"category":36,"text_question":345,"slug":346},534018,"To make brine, José buys 1 kg of salt and pays 12 pesos. If he buys 4 kg, they charge him 48 pesos, but for 100 pesos they sell him 9 kg. What is the constant of proportionality?","to-make-brine-jose-buys-1-kg-of-salt-and-pays-12-pesos-if-he-buys-4-kg-they-charge-him-48-pesos-but-for-100-pesos-they-sell-him-9-kg-what-is-the-constant-of-proportionality",{"id":348,"category":36,"text_question":349,"slug":350},534026,"\"If three wolves catch three rabbits in three hours, how many wolves would it take to catch a hundred rabbits in a hundred hours?\"\n\n The answer is the number of response units.","if-three-wolves-catch-three-rabbits-in-three-hours-how-many-wolves-would-it-take-to-catch-a-hundred-rabbits-in-a-hundred-hours-the-answer-is-the-number-of-response-units",{"id":352,"category":36,"text_question":353,"slug":354},534190,"determine the polynomial F of degree 2 that interpolates. f at points (0;1) (2;5) (4;6). calculate F(0.8).\n Note: Using the polynomial expression with difference operator.","determine-the-polynomial-f-of-degree-2-that-interpolates-f-at-points-0-1-2-5-4-6-calculate-f-0-8-note-using-the-polynomial-expression-with-difference-operator",{"id":356,"category":36,"text_question":357,"slug":358},534313,"The question is using rule 72 determine Kari wants to save 10,000 for a down payment on a house. Illustrate the difference in years it will take her to double her current 5,000 savings based on 6%, 12% and 18% interest rate .","the-question-is-using-rule-72-determine-kari-wants-to-save-10-000-for-a-down-payment-on-a-house-illustrate-the-difference-in-years-it-will-take-her-to-double-her-current-5-000-savings-based-on-6-12",{"id":360,"category":36,"text_question":361,"slug":362},534351,"Fill in the P(X-x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are -5 ,3 , 4, 5 , and 6.","fill-in-the-p-x-x-values-to-give-a-legitimate-probability-distribution-for-the-discrete-random-variable-x-whose-possible-values-are-5-3-4-5-and-6",{"id":364,"category":36,"text_question":365,"slug":366},534413,"15.A newly married couple purchased a home with a $123710 down payment. They financed the remaining balance of the home with a mortgage. Their payments were $15395 at the end of every six months for 23 years and the interest rate was 10.6%, compounded semi-annually. How much did they purchase their home for.\n\nEnter to the nearest cent (two decimals). Do not use $ signs or commas in the answer.","15-a-newly-married-couple-purchased-a-home-with-a-123710-down-payment-they-financed-the-remaining-balance-of-the-home-with-a-mortgage-their-payments-were-15395-at-the-end-of-every-six-months-for-2",{"id":368,"category":36,"text_question":369,"slug":370},534449,"Associate each 2nd degree equation with its respective roots.\n A) x2+6x+8=0\n B)x2-5x-6=0","associate-each-2nd-degree-equation-with-its-respective-roots-a-x2-6x-8-0-b-x2-5x-6-0",{"id":372,"category":36,"text_question":373,"slug":374},534490,"The blood types of individuals in society are as follows: A: 30%, B: 25%, AB: 20%, 0: 25%. It is known that the rates of contracting a certain disease according to blood groups are as follows: A: 7%, B: 6%, AB: 7%, 0: 4%. Accordingly, if a person selected by chance is known to have this disease, what is the probability of having blood group O?","the-blood-types-of-individuals-in-society-are-as-follows-a-30-b-25-ab-20-0-25-it-is-known-that-the-rates-of-contracting-a-certain-disease-according-to-blood-groups-are-as-follows-a-7",{"id":376,"category":36,"text_question":377,"slug":378},534504,"Gender and communication : \nAnswer the question ( 1 paragraph is ok) . Please can you write about women? \n\nCompared to your other identities, how much of a role does gender play in your life? And has your own sex/gender offered you privileges or disadvantages? How so?","gender-and-communication-answer-the-question-1-paragraph-is-ok-please-can-you-write-about-women-compared-to-your-other-identities-how-much-of-a-role-does-gender-play-in-your-life-and-has",{"id":380,"category":36,"text_question":381,"slug":382},534519,"Find I (Intrest) using simple interest formula of 17700 @ 15% for 4 years","find-i-intrest-using-simple-interest-formula-of-17700-15-for-4-years",{"id":384,"category":36,"text_question":385,"slug":386},534552,"Solve for z:\n\n2z-6=10z+2","solve-for-z-2z-6-10z-2",{"id":388,"category":36,"text_question":389,"slug":390},534634,"64-6x^2>0","64-6x-2-0",{"id":392,"category":36,"text_question":393,"slug":394},534693,"Find the number of liters of water needed to reduce 9 liters of lotion.\n shave containing 50% alcohol to a lotion containing 30% alcohol.","find-the-number-of-liters-of-water-needed-to-reduce-9-liters-of-lotion-shave-containing-50-alcohol-to-a-lotion-containing-30-alcohol",{"data":396},[397,401,405],{"id":398,"question":399,"answer":400},145528,"Math question: Find the limit as x approaches 3 of (2x-1)/(x+2).","Answer: Using the properties of limits, we simplify to (2(3)-1)/(3+2) = 5/5 = 1. Therefore, the limit is 1.",{"id":402,"question":403,"answer":404},158795,"What is 3/5 of 80?","To find 3/5 of 80, multiply 80 by 3/5. 80 * 3/5 = 48. Therefore, 3/5 of 80 is 48.",{"id":406,"question":407,"answer":408},144899,"What is the derivative of f(x)=3x^2 + sin(x) at x=π/4?","The derivative of f(x) is f'(x) = 6x + cos(x). Evaluating at x=π/4, f'(π/4) ≈ 2.536.",{"$sicons":410},{"bxl:facebook-circle":411,"bxl:instagram":415,"mdi:web":417,"la:apple":419,"ph:google-logo-bold":422,"ph:google-logo":425},{"left":412,"top":412,"width":413,"height":413,"rotate":412,"vFlip":185,"hFlip":185,"body":414},0,24,"\u003Cpath fill=\"currentColor\" d=\"M12.001 2.002c-5.522 0-9.999 4.477-9.999 9.999c0 4.99 3.656 9.126 8.437 9.879v-6.988h-2.54v-2.891h2.54V9.798c0-2.508 1.493-3.891 3.776-3.891c1.094 0 2.24.195 2.24.195v2.459h-1.264c-1.24 0-1.628.772-1.628 1.563v1.875h2.771l-.443 2.891h-2.328v6.988C18.344 21.129 22 16.992 22 12.001c0-5.522-4.477-9.999-9.999-9.999\"/>",{"left":412,"top":412,"width":413,"height":413,"rotate":412,"vFlip":185,"hFlip":185,"body":416},"\u003Cpath fill=\"currentColor\" d=\"M11.999 7.377a4.623 4.623 0 1 0 0 9.248a4.623 4.623 0 0 0 0-9.248m0 7.627a3.004 3.004 0 1 1 0-6.008a3.004 3.004 0 0 1 0 6.008\"/>\u003Ccircle cx=\"16.806\" cy=\"7.207\" r=\"1.078\" fill=\"currentColor\"/>\u003Cpath fill=\"currentColor\" d=\"M20.533 6.111A4.6 4.6 0 0 0 17.9 3.479a6.6 6.6 0 0 0-2.186-.42c-.963-.042-1.268-.054-3.71-.054s-2.755 0-3.71.054a6.6 6.6 0 0 0-2.184.42a4.6 4.6 0 0 0-2.633 2.632a6.6 6.6 0 0 0-.419 2.186c-.043.962-.056 1.267-.056 3.71s0 2.753.056 3.71c.015.748.156 1.486.419 2.187a4.6 4.6 0 0 0 2.634 2.632a6.6 6.6 0 0 0 2.185.45c.963.042 1.268.055 3.71.055s2.755 0 3.71-.055a6.6 6.6 0 0 0 2.186-.419a4.6 4.6 0 0 0 2.633-2.633c.263-.7.404-1.438.419-2.186c.043-.962.056-1.267.056-3.71s0-2.753-.056-3.71a6.6 6.6 0 0 0-.421-2.217m-1.218 9.532a5 5 0 0 1-.311 1.688a3 3 0 0 1-1.712 1.711a5 5 0 0 1-1.67.311c-.95.044-1.218.055-3.654.055c-2.438 0-2.687 0-3.655-.055a5 5 0 0 1-1.669-.311a3 3 0 0 1-1.719-1.711a5.1 5.1 0 0 1-.311-1.669c-.043-.95-.053-1.218-.053-3.654s0-2.686.053-3.655a5 5 0 0 1 .311-1.687c.305-.789.93-1.41 1.719-1.712a5 5 0 0 1 1.669-.311c.951-.043 1.218-.055 3.655-.055s2.687 0 3.654.055a5 5 0 0 1 1.67.311a3 3 0 0 1 1.712 1.712a5.1 5.1 0 0 1 .311 1.669c.043.951.054 1.218.054 3.655s0 2.698-.043 3.654z\"/>",{"left":412,"top":412,"width":413,"height":413,"rotate":412,"vFlip":185,"hFlip":185,"body":418},"\u003Cpath fill=\"currentColor\" d=\"M16.36 14c.08-.66.14-1.32.14-2s-.06-1.34-.14-2h3.38c.16.64.26 1.31.26 2s-.1 1.36-.26 2m-5.15 5.56c.6-1.11 1.06-2.31 1.38-3.56h2.95a8.03 8.03 0 0 1-4.33 3.56M14.34 14H9.66c-.1-.66-.16-1.32-.16-2s.06-1.35.16-2h4.68c.09.65.16 1.32.16 2s-.07 1.34-.16 2M12 19.96c-.83-1.2-1.5-2.53-1.91-3.96h3.82c-.41 1.43-1.08 2.76-1.91 3.96M8 8H5.08A7.92 7.92 0 0 1 9.4 4.44C8.8 5.55 8.35 6.75 8 8m-2.92 8H8c.35 1.25.8 2.45 1.4 3.56A8 8 0 0 1 5.08 16m-.82-2C4.1 13.36 4 12.69 4 12s.1-1.36.26-2h3.38c-.08.66-.14 1.32-.14 2s.06 1.34.14 2M12 4.03c.83 1.2 1.5 2.54 1.91 3.97h-3.82c.41-1.43 1.08-2.77 1.91-3.97M18.92 8h-2.95a15.7 15.7 0 0 0-1.38-3.56c1.84.63 3.37 1.9 4.33 3.56M12 2C6.47 2 2 6.5 2 12a10 10 0 0 0 10 10a10 10 0 0 0 10-10A10 10 0 0 0 12 2\"/>",{"left":412,"top":412,"width":420,"height":420,"rotate":412,"vFlip":185,"hFlip":185,"body":421},32,"\u003Cpath fill=\"currentColor\" d=\"M20.844 2c-1.64 0-3.297.852-4.407 2.156v.032c-.789.98-1.644 2.527-1.375 4.312c-.128-.05-.136-.035-.28-.094c-.692-.281-1.548-.594-2.563-.594c-3.98 0-7 3.606-7 8.344c0 3.067 1.031 5.942 2.406 8.094c.688 1.078 1.469 1.965 2.281 2.625S11.57 28 12.531 28s1.68-.324 2.219-.563c.54-.238.957-.437 1.75-.437c.715 0 1.078.195 1.625.438c.547.242 1.293.562 2.281.562c1.07 0 1.98-.523 2.719-1.188s1.36-1.519 1.875-2.343c.516-.824.922-1.633 1.219-2.282c.148-.324.258-.593.343-.812s.13-.281.188-.531l.188-.813l-.75-.343a5.3 5.3 0 0 1-1.5-1.063c-.625-.637-1.157-1.508-1.157-2.844A4.08 4.08 0 0 1 24.563 13c.265-.309.542-.563.75-.719c.105-.078.187-.117.25-.156c.062-.04.05-.027.156-.094l.843-.531l-.562-.844c-1.633-2.511-4.246-2.844-5.281-2.844c-.48 0-.82.168-1.25.25c.242-.226.554-.367.75-.624c.004-.004-.004-.028 0-.032q.018-.016.031-.031h.031a6.16 6.16 0 0 0 1.563-4.438L21.78 2zm-1.188 2.313c-.172.66-.453 1.289-.906 1.78l-.063.063c-.382.516-.972.899-1.562 1.125c.164-.652.45-1.312.844-1.812c.008-.012.023-.02.031-.032c.438-.5 1.043-.875 1.656-1.125zm-7.437 5.5c.558 0 1.172.21 1.812.468s1.239.594 2.094.594c.852 0 1.496-.336 2.25-.594s1.559-.469 2.344-.469c.523 0 1.816.333 2.906 1.344c-.191.172-.36.297-.563.531a6.2 6.2 0 0 0-1.53 4.094c0 1.906.831 3.34 1.718 4.25c.55.563.89.696 1.313.938c-.055.125-.086.222-.157.375a19 19 0 0 1-1.093 2.062c-.454.727-1.004 1.434-1.532 1.907c-.527.472-1 .687-1.375.687c-.566 0-.898-.156-1.468-.406S17.581 25 16.5 25c-1.137 0-1.977.336-2.563.594c-.585.258-.89.406-1.406.406c-.246 0-.777-.2-1.375-.688c-.597-.488-1.254-1.23-1.844-2.156c-1.183-1.851-2.093-4.394-2.093-7c0-3.941 2.199-6.343 5-6.343\"/>",{"left":412,"top":412,"width":423,"height":423,"rotate":412,"vFlip":185,"hFlip":185,"body":424},256,"\u003Cpath fill=\"currentColor\" d=\"M228 128a100 100 0 1 1-22.86-63.64a12 12 0 0 1-18.51 15.28A76 76 0 1 0 203.05 140H128a12 12 0 0 1 0-24h88a12 12 0 0 1 12 12\"/>",{"left":412,"top":412,"width":423,"height":423,"rotate":412,"vFlip":185,"hFlip":185,"body":426},"\u003Cpath fill=\"currentColor\" d=\"M224 128a96 96 0 1 1-21.95-61.09a8 8 0 1 1-12.33 10.18A80 80 0 1 0 207.6 136H128a8 8 0 0 1 0-16h88a8 8 0 0 1 8 8\"/>",{"oVhJaef6Ht":8,"t96FybqVTi":8,"5lK7LS5al0":8,"5oSQ2a90xd":8,"P8SGtqjq9X":8,"2QISyIzlyM":8,"HGsO2Ckakl":8},"/general/two-numbers-differ-by-7-and-the-sum-of-their-squares-is-29-find-the-numbers"] AppleWebKit/537.36 $KHTML, like Gecko$ Chrome/64.0.3282.39 Safari/537.36",refreshOnResize:false}},app:{baseURL:"/",buildAssetsDir:"/_nuxt/",cdnURL:"https://gcdn.fx2.io/math-master.org/"}}