:\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":43,"category":28,"text_question":44,"photo_question":30,"text_answer":45,"step_text_answer":8,"step_photo_answer":8,"views":46,"likes":47,"slug":48},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":50,"category":28,"text_question":51,"photo_question":30,"text_answer":52,"step_text_answer":8,"step_photo_answer":8,"views":53,"likes":54,"slug":55},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":57,"category":28,"text_question":58,"photo_question":30,"text_answer":59,"step_text_answer":8,"step_photo_answer":8,"views":60,"likes":61,"slug":62},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":64,"category":28,"text_question":65,"photo_question":30,"text_answer":66,"step_text_answer":8,"step_photo_answer":8,"views":67,"likes":68,"slug":69},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":71,"category":28,"text_question":72,"photo_question":30,"text_answer":73,"step_text_answer":8,"step_photo_answer":8,"views":74,"likes":75,"slug":76},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":78,"category":28,"text_question":79,"photo_question":30,"text_answer":80,"step_text_answer":8,"step_photo_answer":8,"views":81,"likes":82,"slug":83},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":85,"category":28,"text_question":86,"photo_question":30,"text_answer":87,"step_text_answer":8,"step_photo_answer":8,"views":88,"likes":89,"slug":90},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":92,"category":28,"text_question":93,"photo_question":30,"text_answer":94,"step_text_answer":8,"step_photo_answer":8,"views":95,"likes":96,"slug":97},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":99,"category":28,"text_question":100,"photo_question":30,"text_answer":101,"step_text_answer":8,"step_photo_answer":8,"views":102,"likes":103,"slug":104},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":106,"category":28,"text_question":107,"photo_question":30,"text_answer":108,"step_text_answer":8,"step_photo_answer":8,"views":109,"likes":110,"slug":111},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":113,"category":28,"text_question":114,"photo_question":30,"text_answer":115,"step_text_answer":8,"step_photo_answer":8,"views":116,"likes":117,"slug":118},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":120,"category":28,"text_question":121,"photo_question":30,"text_answer":122,"step_text_answer":8,"step_photo_answer":8,"views":123,"likes":124,"slug":125},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":127,"category":28,"text_question":128,"photo_question":30,"text_answer":129,"step_text_answer":8,"step_photo_answer":8,"views":130,"likes":131,"slug":132},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":134,"category":28,"text_question":135,"photo_question":30,"text_answer":136,"step_text_answer":8,"step_photo_answer":8,"views":137,"likes":138,"slug":139},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":141,"category":28,"text_question":142,"photo_question":30,"text_answer":143,"step_text_answer":8,"step_photo_answer":8,"views":144,"likes":145,"slug":146},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":148,"category":28,"text_question":149,"photo_question":30,"text_answer":150,"step_text_answer":8,"step_photo_answer":8,"views":151,"likes":152,"slug":153},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":155,"category":28,"text_question":156,"photo_question":30,"text_answer":157,"step_text_answer":8,"step_photo_answer":8,"views":158,"likes":159,"slug":160},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":162,"category":28,"text_question":163,"photo_question":30,"text_answer":164,"step_text_answer":8,"step_photo_answer":8,"views":165,"likes":166,"slug":167},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":169,"prev":8,"next":10},186,{"current_page":6,"from":6,"last_page":169,"links":171,"path":205,"per_page":206,"to":206,"total":207},[172,175,178,180,182,184,186,189,192,195,198,201,203],{"url":6,"label":173,"active":174},"1",true,{"url":10,"label":176,"active":177},"2",false,{"url":13,"label":179,"active":177},"3",{"url":16,"label":181,"active":177},"4",{"url":19,"label":183,"active":177},"5",{"url":22,"label":185,"active":177},"6",{"url":187,"label":188,"active":177},7,"7",{"url":190,"label":191,"active":177},8,"8",{"url":193,"label":194,"active":177},9,"9",{"url":196,"label":197,"active":177},10,"10",{"url":199,"label":200,"active":177},185,"185",{"url":169,"label":202,"active":177},"186",{"url":10,"label":204,"active":177},"Next »","https://api.math-master.org/api/question",20,3704,{"data":209},[210,211,212,213,214,215],{"id":6,"title":7,"slug":8},{"id":10,"title":11,"slug":8},{"id":13,"title":14,"slug":8},{"id":16,"title":17,"slug":8},{"id":19,"title":20,"slug":8},{"id":22,"title":23,"slug":8},{"data":217},{"id":218,"category":28,"slug":219,"text_question":220,"photo_question":8,"text_answer":221,"step_text_answer":8,"step_photo_answer":8,"views":222,"likes":223,"expert":224},537593,"show-24x25-in-distributive-property","Show 24x25 in Distributive property","1. Use the distributive property: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>24 \\times 25 = 24 \\times (20 + 5)\u003C/math-field>\u003C/math-field> \u003Cbr />\n2. Apply the distributive property: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>24 \\times 20 + 24 \\times 5\u003C/math-field>\u003C/math-field> \u003Cbr />\n3. Calculate: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>24 \\times 20 = 480\u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>24 \\times 5 = 120\u003C/math-field>\u003C/math-field> \u003Cbr />\n4. Add the results: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>480 + 120 = 600\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\nThe answer is: 600",1125,225,{"id":225,"name":226,"photo":227,"biography":228,"created_at":8,"updated_at":8,"rating":229,"total_answer":230},38,"Andrea","https://api.math-master.org/img/experts/38/38.webp","I became interested in mathematics in school. Took part in Olympiads and scientific competitions. In 2010, entered at Taras Shevchenko National University of Kyiv, Faculty of Mechanics and Mathematics. In 2021, conferred the Philosophy Doctor Degree Field of Study Mathematics and statistics Progranime Subject Area Mathematics. I am the author of 5 scientific articles. I have also been working as a mathematics reactor for more than 10 years.",4.5,80,{"data":232},{"questions":233},[234,238,242,246,250,254,258,262,266,270,274,278,282,286,290,294,298,302,306,310],{"id":235,"category":28,"text_question":236,"slug":237},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":239,"category":28,"text_question":240,"slug":241},532088,"1/2x +3 \u003C4x-7","1-2x-3-4x-7",{"id":243,"category":28,"text_question":244,"slug":245},533940,"4.2x10^_6 convert to standard notation","4-2x10-6-convert-to-standard-notation",{"id":247,"category":28,"text_question":248,"slug":249},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",{"id":251,"category":28,"text_question":252,"slug":253},534063,"A National Solidarity Bond offers A 5 year bond offering a gross return of 15%\r\n\r\nCalculate the AER for this investment. (Give your answer to two decimal places, no need for the percent or € sign in your answer)","a-national-solidarity-bond-offers-a-5-year-bond-offering-a-gross-return-of-15-calculate-the-aer-for-this-investment-give-your-answer-to-two-decimal-places-no-need-for-the-percent-or-sign-in",{"id":255,"category":28,"text_question":256,"slug":257},534076,"4. Show that if n is any integer, then n^2 3n 5 is an odd integer","4-show-that-if-n-is-any-integer-then-n-2-3n-5-is-an-odd-integer",{"id":259,"category":28,"text_question":260,"slug":261},534113,"There are four times as many roses as tulips in Claire’s garden. Claire picked half of the number of roses and 140 roses were left in the garden. How many roses and tulips were in the Garden the first?","there-are-four-times-as-many-roses-as-tulips-in-claire-s-garden-claire-picked-half-of-the-number-of-roses-and-140-roses-were-left-in-the-garden-how-many-roses-and-tulips-were-in-the-garden-the-first",{"id":263,"category":28,"text_question":264,"slug":265},534174,"find f(x) for f'(x)=3x+7","find-f-x-for-f-x-3x-7",{"id":267,"category":28,"text_question":268,"slug":269},534230,"User\nBefore the election, a poll of 60 voters found the proportion who support the Green candidate to be 25%. Calculate the 90% confidence interval for the population parameter. (Give your answers as a PERCENTAGE rounded to TWO DECIMAL PLACES: exclude any trailing zeros and DO NOT INSERT THE % SIGN)\nGive the lower limit of the 90% confidence interval \nGive the upper limit of the 90% confidence interval","user-before-the-election-a-poll-of-60-voters-found-the-proportion-who-support-the-green-candidate-to-be-25-calculate-the-90-confidence-interval-for-the-population-parameter-give-your-answers-as",{"id":271,"category":28,"text_question":272,"slug":273},534258,"Shows two blocks, masses 4.3 kg and 5.4 kg, being pushed across a frictionless surface by a 22.5-N horizontal force applied to the 4.3-kg block.\nA. What is the acceleration of the blocks?\nB. What is the force of the 4.3-kg block on the 5.4 -kg block?\nC. What is the force of the 5.4 -kg block on the 4.3 -kg block?","shows-two-blocks-masses-4-3-kg-and-5-4-kg-being-pushed-across-a-frictionless-surface-by-a-22-5-n-horizontal-force-applied-to-the-4-3-kg-block-a-what-is-the-acceleration-of-the-blocks-b-what-is-t",{"id":275,"category":28,"text_question":276,"slug":277},534305,"User\n One of the applications of the derivative of a function is its use in Physics, where a function that at every instant t associates the number s(t), this function s is called the clockwise function of the movement.\n\n By deriving the time function we obtain the velocity function at time t, denoted by v(t).\n A body has a time function that determines its position in meters at time t as S(t)=t.³√t+2.t . Present the speed of this body at time t = 8 s.","user-one-of-the-applications-of-the-derivative-of-a-function-is-its-use-in-physics-where-a-function-that-at-every-instant-t-associates-the-number-s-t-this-function-s-is-called-the-clockwise-functi",{"id":279,"category":28,"text_question":280,"slug":281},534356,"Find the complement and supplement angles of 73","find-the-complement-and-supplement-angles-of-73",{"id":283,"category":28,"text_question":284,"slug":285},534388,"A contractor gives a bank note for $10250 at a rate of 1% for one month. How much interest\nis charged for 4 months?","a-contractor-gives-a-bank-note-for-10250-at-a-rate-of-1-for-one-month-how-much-interest-is-charged-for-4-months",{"id":287,"category":28,"text_question":288,"slug":289},534446,"To get to a hotel on the hill you have to travel 6 km of uphill road and every kilometer there are 6 sharp curves. Each of the sharp curves is marked by three traffic signs. How many traffic signs are there on the stretch of road that leads to the arbergi?","to-get-to-a-hotel-on-the-hill-you-have-to-travel-6-km-of-uphill-road-and-every-kilometer-there-are-6-sharp-curves-each-of-the-sharp-curves-is-marked-by-three-traffic-signs-how-many-traffic-signs-are",{"id":291,"category":28,"text_question":292,"slug":293},534453,"A person runs 175 yards per minute write a variable that represents the relationship between time and distance","a-person-runs-175-yards-per-minute-write-a-variable-that-represents-the-relationship-between-time-and-distance",{"id":295,"category":28,"text_question":296,"slug":297},534546,"The mean of 4 numbers is 5 and the mean of 3 different numbers is 12. What is the mean of the 7 numbers together? Produce an algebraic solution. Guess and check is acceptable.","the-mean-of-4-numbers-is-5-and-the-mean-of-3-different-numbers-is-12-what-is-the-mean-of-the-7-numbers-together-produce-an-algebraic-solution-guess-and-check-is-acceptable",{"id":299,"category":28,"text_question":300,"slug":301},534553,"2x-4=8","2x-4-8",{"id":303,"category":28,"text_question":304,"slug":305},534593,"Determine the general solution of the equation y′+y=e−x\n .","determine-the-general-solution-of-the-equation-y-y-e-x",{"id":307,"category":28,"text_question":308,"slug":309},534679,"Find the rule that connects the first number to the second number of each pair.\r\nApply the rule to find the missing number in the third pair.\r\n(18 is to 22) (54 is to 26) (9 is to ?)","find-the-rule-that-connects-the-first-number-to-the-second-number-of-each-pair-apply-the-rule-to-find-the-missing-number-in-the-third-pair-18-is-to-22-54-is-to-26-9-is-to",{"id":311,"category":28,"text_question":312,"slug":313},534691,"5 1/9 + 2 2/3","5-1-9-2-2-3",{"data":315},{"questions":316},[317,321,325,329,333,337,341,345,349,353,357,361,365,369,373,377,381,385,389,393],{"id":318,"category":28,"text_question":319,"slug":320},532035,"I want to divide R$ 2200.00 between Antônio, Beto and Cássia, so that Beto receives half from Antônio and Cássia receives a third of Beto. Under these conditions, how much more will Beto receive than Cássia?","i-want-to-divide-r-2200-00-between-antonio-beto-and-cassia-so-that-beto-receives-half-from-antonio-and-cassia-receives-a-third-of-beto-under-these-conditions-how-much-more-will-beto-receive-than",{"id":322,"category":28,"text_question":323,"slug":324},532045,"5 . {2/5 + [ (8/-9) - (1/-7) + (-2/5) ] ÷ (2/-5)} . 8/15","5-2-5-8-9-1-7-2-5-2-5-8-15",{"id":326,"category":28,"text_question":327,"slug":328},532046,"If you have a bag with 18 white balls and 2 black balls. What is the probability of drawing a white ball? And extracting a black one?","if-you-have-a-bag-with-18-white-balls-and-2-black-balls-what-is-the-probability-of-drawing-a-white-ball-and-extracting-a-black-one",{"id":330,"category":28,"text_question":331,"slug":332},532052,"-6n+5=-13","6n-5-13",{"id":334,"category":28,"text_question":335,"slug":336},532059,"-6(3x-4)=-6","6-3x-4-6",{"id":338,"category":28,"text_question":339,"slug":340},534002,"Equivalent expression of the sequence (3n-4)-(n-2)","equivalent-expression-of-the-sequence-3n-4-n-2",{"id":342,"category":28,"text_question":343,"slug":344},534170,"Find all real numbers x that satisfy the equation \\sqrt{x^2-2}=\\sqrt{3-x}","find-all-real-numbers-x-that-satisfy-the-equation-sqrt-x-2-2-sqrt-3-x",{"id":346,"category":28,"text_question":347,"slug":348},534173,"sin 30","sin-30",{"id":350,"category":28,"text_question":351,"slug":352},534191,"6-35 A recent study by an environmental watchdog determined that the amount of contaminants\n in Minnesota lakes (in parts per million) it has a normal distribution with a mean of 64 ppm and variance of 17.6. Assume that 35 lakes are randomly selected and sampled. Find the probability that the sample average of the amount of contaminants is\n a) Greater than 72 ppm.\n b) Between 64 and 72 ppm.\n c) Exactly 64 ppm.\n d) Greater than 94 ppm.","6-35-a-recent-study-by-an-environmental-watchdog-determined-that-the-amount-of-contaminants-in-minnesota-lakes-in-parts-per-million-it-has-a-normal-distribution-with-a-mean-of-64-ppm-and-variance-o",{"id":354,"category":28,"text_question":355,"slug":356},534275,"3/9*4/8=","3-9-4-8",{"id":358,"category":28,"text_question":359,"slug":360},534286,"From 1975 through 2020 the mean annual gain of the Dow Jones Industrial Average was 652. A random sample of 34 years is selected from this population. What is the probability that the mean gain for the sample was between 400 and 800? Assume the standard deviation is 1539","from-1975-through-2020-the-mean-annual-gain-of-the-dow-jones-industrial-average-was-652-a-random-sample-of-34-years-is-selected-from-this-population-what-is-the-probability-that-the-mean-gain-for-th",{"id":362,"category":28,"text_question":363,"slug":364},534339,"X~N(2.6,1.44). find the P(X\u003C3.1)","x-n-2-6-1-44-find-the-p-x-3-1",{"id":366,"category":28,"text_question":367,"slug":368},534364,"List five numbers that belong to the 5 (mod 6) numbers. Alternate phrasing, list five numbers that satisfy equation x = 5 (mod 6)","list-five-numbers-that-belong-to-the-5-mod-6-numbers-alternate-phrasing-list-five-numbers-that-satisfy-equation-x-5-mod-6",{"id":370,"category":28,"text_question":371,"slug":372},534406,"During a month's time, an automobile sales person receives a 6% commission on the first $5000 in sales, a 7% commission on the next $5000 sales, 8% commission on anything over $10,000. What is her commission for $36,000 in sales?","during-a-month-s-time-an-automobile-sales-person-receives-a-6-commission-on-the-first-5000-in-sales-a-7-commission-on-the-next-5000-sales-8-commission-on-anything-over-10-000-what-is-her-com",{"id":374,"category":28,"text_question":375,"slug":376},534423,"find missing measure for triangle \narea = 48 m square\nbase = 10m\nheaighy = ? m","find-missing-measure-for-triangle-area-48-m-square-base-10m-heaighy-m",{"id":378,"category":28,"text_question":379,"slug":380},534441,"A buyer purchased a North Carolina home for $475,250. The seller allowed the buyer to assume his first small mortgage with a loan balance of $110,000. How much is the excise tax paid in the transaction?\n$951\n$729.50\n$950.50\n$221\nnone of the above","a-buyer-purchased-a-north-carolina-home-for-475-250-the-seller-allowed-the-buyer-to-assume-his-first-small-mortgage-with-a-loan-balance-of-110-000-how-much-is-the-excise-tax-paid-in-the-transactio",{"id":382,"category":28,"text_question":383,"slug":384},534451,"Pablo has a balance of $440,000 and 2/5 of the money is used to pay bills. How much money do you have left after paying the bills?","pablo-has-a-balance-of-440-000-and-2-5-of-the-money-is-used-to-pay-bills-how-much-money-do-you-have-left-after-paying-the-bills",{"id":386,"category":28,"text_question":387,"slug":388},534505,"How do you convert a fraction to a decimal","how-do-you-convert-a-fraction-to-a-decimal",{"id":390,"category":28,"text_question":391,"slug":392},534641,"3(x-4)=156","3-x-4-156",{"id":394,"category":28,"text_question":395,"slug":396},534661,"Dano forgot his computer password. The password was four characters long. Dano remembered only three characters: 3, g, N. The last character was one of the numbers 3, 5, 7, 9. How many possible expansions are there for Dano's password?","dano-forgot-his-computer-password-the-password-was-four-characters-long-dano-remembered-only-three-characters-3-g-n-the-last-character-was-one-of-the-numbers-3-5-7-9-how-many-possible-expans",{"data":398},[399,403,407],{"id":400,"question":401,"answer":402},105205,"Math question: \"Factor the expression 4x^2 + 12x + 9 using the quadratic factoring formula.\"","Answer: To factorize 4x^2 + 12x + 9, we use the quadratic factoring formula. The equation can be written as (2x + 3)^2 which is the square of the binomial 2x + 3.",{"id":404,"question":405,"answer":406},121890,"What is the equation of an ellipse with a center at (4, 2), major axis length of 10, and minor axis length of 6?","The equation of the ellipse is (x-4)^2/25 + (y-2)^2/9 = 1. The major axis is horizontal, centered at (4,2), with a length of 10 units, and the minor axis is vertical, centered at (4,2), with a length of 6 units.",{"id":408,"question":409,"answer":410},120405,"Find the volume of a sphere with a radius of 5cm.","The volume V of a sphere with radius r is given by V = (4/3)πr³. Substituting r = 5, we have V = (4/3)π(5)³ = 523.6 cm³.",{"$sicons":412},{"bxl:facebook-circle":413,"bxl:instagram":417,"mdi:web":419,"la:apple":421,"ph:google-logo-bold":424,"ph:google-logo":427},{"left":414,"top":414,"width":415,"height":415,"rotate":414,"vFlip":177,"hFlip":177,"body":416},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":414,"top":414,"width":415,"height":415,"rotate":414,"vFlip":177,"hFlip":177,"body":418},"\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\" 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