:\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},532310,"the-random-variable-y-is-defined-as-the-sum-between-two-different-integers-selected-at-random-between-4-and-2-both-included-what-are-the-possible-values-of-the-random-variable-y-what-is-the-value","The random variable Y is defined as the sum between two different integers selected at random between -4 and 2 (both included). What are the possible values of the random variable Y? What is the value of P(Y=-3)? And whether it is less than or equal to -5?","The possible numbers are -4,-3,-2,-1,0,1,2\nP(Y=-3)=1/7",279,56,{"id":225,"name":226,"photo":227,"biography":228,"created_at":8,"updated_at":8,"rating":229,"total_answer":124},11,"Lurline","https://api.math-master.org/img/experts/11/11.webp","Hello there! I am Aditi Kumari. I am from India and currently I am pursuing my masters degree in the field of economics from the Delhi School of Economics - one of the topmost institution. I have completed my bachelors in economics from Shri Ram College of Commerce which is the top ranked college in India for the study of economics.\nEconomics is a study of application of mathematics on social science. since past four years I would say I have learned in depth application of math in the field of economics. It includes using advance calculus, linear algebra, statistics, econometrics, real analysis, and optimization. I was amazed initially that we can apply so much maths to social science subject. Before coming to university life I had no idea that for the next few years all I will be practicing is maths. Calculating marginal unitity, demand behaviour, doing differentiation, integration, algebra, etc. But having said that I have never got bored of the subject, every new topic opens up new dimension of the subject and you have to start thinking afresh. I would say it is this exciting nature of the subject which made it my favourite subject in school. I always wanted to keep studying maths and keep it as my bestie. The love for the subject was there since the first grade and continues till now.\nMaths was one subject in which I mostly scored the highest, but more than that it's the nature of the subject and insightful results which made me practice it again and again and never let me confine to just a single book.",4.6,{"data":231},{"questions":232},[233,237,241,245,249,252,256,260,264,268,272,276,280,284,288,292,295,299,303,307],{"id":234,"category":28,"text_question":235,"slug":236},532053,"Y=-x^2-8x-15 X=-7","y-x-2-8x-15-x-7",{"id":238,"category":28,"text_question":239,"slug":240},532079,"The gross domestic product the gdp for the United States in 2017 was approximately $2.05x10^3. If you wrote this number in standard notation , it would be 205 followed by how many zeros","the-gross-domestic-product-the-gdp-for-the-united-states-in-2017-was-approximately-2-05x10-3-if-you-wrote-this-number-in-standard-notation-it-would-be-205-followed-by-how-many-zeros",{"id":242,"category":28,"text_question":243,"slug":244},532086,"-8+3/5","8-3-5",{"id":246,"category":28,"text_question":247,"slug":248},532094,"Karina has a plot of 5,000 square meters in which she has decided that 60% of it will be used to plant vegetables. Of this part, 12% will be dedicated to planting lettuce. How much surface area of the plot will be used for cultivation?","karina-has-a-plot-of-5-000-square-meters-in-which-she-has-decided-that-60-of-it-will-be-used-to-plant-vegetables-of-this-part-12-will-be-dedicated-to-planting-lettuce-how-much-surface-area-of-the",{"id":250,"category":28,"text_question":251,"slug":251},533975,"4x567",{"id":253,"category":28,"text_question":254,"slug":255},533977,"what is the annual rate on ​$525 at 0.046​% per day for 3 months?","what-is-the-annual-rate-on-525-at-0-046-per-day-for-3-months",{"id":257,"category":28,"text_question":258,"slug":259},534031,"3(2•1+3)4","3-2-1-3-4",{"id":261,"category":28,"text_question":262,"slug":263},534057,"Identify a pattern in the list of numbers.Then use this pattern to find the next number. 37,31,25,19,13","identify-a-pattern-in-the-list-of-numbers-then-use-this-pattern-to-find-the-next-number-37-31-25-19-13",{"id":265,"category":28,"text_question":266,"slug":267},534071,"There are 162 students enrolled in the basic mathematics course. If the number of women is 8 times the number of men, how many women are there in the basic mathematics course?","there-are-162-students-enrolled-in-the-basic-mathematics-course-if-the-number-of-women-is-8-times-the-number-of-men-how-many-women-are-there-in-the-basic-mathematics-course",{"id":269,"category":28,"text_question":270,"slug":271},534121,"In a grocery store, when you take out 3 peppers and 4 carrots, there are 26 peppers and 46 carrots left.\n How many peppers and carrots were there initially?","in-a-grocery-store-when-you-take-out-3-peppers-and-4-carrots-there-are-26-peppers-and-46-carrots-left-how-many-peppers-and-carrots-were-there-initially",{"id":273,"category":28,"text_question":274,"slug":275},534126,"∫ √9x + 1 dx","9x-1-dx",{"id":277,"category":28,"text_question":278,"slug":279},534178,"Two business partners have a bank balance of $17,942.00. After the first year their interest brings their balance to $18,928.91. What rate of interest is earned?","two-business-partners-have-a-bank-balance-of-17-942-00-after-the-first-year-their-interest-brings-their-balance-to-18-928-91-what-rate-of-interest-is-earned",{"id":281,"category":28,"text_question":282,"slug":283},534246,"How to do 15 x 3304","how-to-do-15-x-3304",{"id":285,"category":28,"text_question":286,"slug":287},534361,"cube root of 56","cube-root-of-56",{"id":289,"category":28,"text_question":290,"slug":291},534399,"viii. An ac circuit with a 80 μF capacitor in series with a coil of resistance 16Ω and inductance 160mH is connected to a 100V, 100 Hz supply is shown below. Calculate\r\n\r\n 7. the inductive reactance \t\r\n 8. the capacitive reactance \t\r\n 9. the circuit impedance and V-I phase angle θ \r\n 10. the circuit current I \t \r\n 11. the phasor voltages VR, VL, VC and VS \t\r\n 12. the resonance circuit frequency \t\r\n\r\nAlso construct a fully labeled and appropriately ‘scaled’ voltage phasor diagram.","viii-an-ac-circuit-with-a-80-uf-capacitor-in-series-with-a-coil-of-resistance-16-and-inductance-160mh-is-connected-to-a-100v-100-hz-supply-is-shown-below-calculate-7-the-inductive-reactanc",{"id":293,"category":28,"text_question":294,"slug":294},534401,"2x2",{"id":296,"category":28,"text_question":297,"slug":298},534511,"A post office has three categories of letters: 60% are from businesses, 30% are individual mail, and the remaining 10% are government mail.\n\n5% of the letters from businesses have address errors, 10% of the individual mail has address errors, while 1% of the government mail has address errors.\n\nIf we receive a letter with an address error, what is the probability that it is individual mail?\"","a-post-office-has-three-categories-of-letters-60-are-from-businesses-30-are-individual-mail-and-the-remaining-10-are-government-mail-5-of-the-letters-from-businesses-have-address-errors-10",{"id":300,"category":28,"text_question":301,"slug":302},534618,"question 1\n Consider a sample space S, and two events A and B such that\n P(A ∩ B) = 0.2, P(A ∪ B) = 0.6, P(B ∪ ̄A) = 0.8\n (a) [0.5 points] Calculate P (A).\n (b) [0.5 points] Calculate P (B)","question-1-consider-a-sample-space-s-and-two-events-a-and-b-such-that-p-a-b-0-2-p-a-b-0-6-p-b-a-0-8-a-0-5-points-calculate-p-a-b-0-5-points-calculate-p-b",{"id":304,"category":28,"text_question":305,"slug":306},534685,"A gas is leaking at 3.5ft3/min in a room of 2.9m by 6.9ft by 15.7m. How long would it take (in seconds) for 22% of the room to reach the LFL, if the gas has a LFL of 2.51%?","a-gas-is-leaking-at-3-5ft3-min-in-a-room-of-2-9m-by-6-9ft-by-15-7m-how-long-would-it-take-in-seconds-for-22-of-the-room-to-reach-the-lfl-if-the-gas-has-a-lfl-of-2-51",{"id":308,"category":28,"text_question":309,"slug":310},534694,"An invoice for €2,880 plus default interest of €48.40 was paid on October 28th. Interest rate 5.5%. When was the bill due?","an-invoice-for-2-880-plus-default-interest-of-48-40-was-paid-on-october-28th-interest-rate-5-5-when-was-the-bill-due",{"data":312},{"questions":313},[314,318,322,326,330,334,338,342,346,350,354,358,362,363,367,371,375,379,383,387],{"id":315,"category":28,"text_question":316,"slug":317},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":319,"category":28,"text_question":320,"slug":321},532072,"Since one of the three integers whose product is (-60) is (+4), write the values that two integers can take.","since-one-of-the-three-integers-whose-product-is-60-is-4-write-the-values-that-two-integers-can-take",{"id":323,"category":28,"text_question":324,"slug":325},532076,"(x^2+3x)/(x^2-9)=","x-2-3x-x-2-9",{"id":327,"category":28,"text_question":328,"slug":329},532089,"Jose bought 3/4 of oil and his sister bought 6/8, which of the two bought more oil?","jose-bought-3-4-of-oil-and-his-sister-bought-6-8-which-of-the-two-bought-more-oil",{"id":331,"category":28,"text_question":332,"slug":333},533896,"What payment 7 months from now would be equivalent in value to a $3,300 payment due 23 months from now? The value of money is 2.7% simple interest. Round your answer to 2 decimal places.\nShow all work and how you arrive at the answer..","what-payment-7-months-from-now-would-be-equivalent-in-value-to-a-3-300-payment-due-23-months-from-now-the-value-of-money-is-2-7-simple-interest-round-your-answer-to-2-decimal-places-show-all-work",{"id":335,"category":28,"text_question":336,"slug":337},533907,"X^2 = 25","x-2-25",{"id":339,"category":28,"text_question":340,"slug":341},533979,"A bird randomly chooses to land on 1 of 12 perches available in its aviary. Determine the Probability of it landing on a perch numbered 8 and then on a perch marked with a prime number; take into account that he never lands on the same perch in the sequence.","a-bird-randomly-chooses-to-land-on-1-of-12-perches-available-in-its-aviary-determine-the-probability-of-it-landing-on-a-perch-numbered-8-and-then-on-a-perch-marked-with-a-prime-number-take-into-acco",{"id":343,"category":28,"text_question":344,"slug":345},533999,"2x-4y=-6; -4y+4y=-8","2x-4y-6-4y-4y-8",{"id":347,"category":28,"text_question":348,"slug":349},534046,"Sean must chose a 6- digit PIN number for his online banking account.Each digit can be chosen from 0 to 9. How many different possible PIN numbers can sean chose.","sean-must-chose-a-6-digit-pin-number-for-his-online-banking-account-each-digit-can-be-chosen-from-0-to-9-how-many-different-possible-pin-numbers-can-sean-chose",{"id":351,"category":28,"text_question":352,"slug":353},534078,"is the x element (180,270), if tanx-3cotx=2, sinx ?","is-the-x-element-180-270-if-tanx-3cotx-2-sinx",{"id":355,"category":28,"text_question":356,"slug":357},534124,"Substitute a=2 and b=-3 and c=-4 to evaluate 2ac/(-2b^2-a)","substitute-a-2-and-b-3-and-c-4-to-evaluate-2ac-2b-2-a",{"id":359,"category":28,"text_question":360,"slug":361},534196,". What will be the osmotic pressure of a solution that was prepared at 91°F by dissolving 534 grams of aluminum hydroxide in enough water to generate 2.784 ml of solution.","what-will-be-the-osmotic-pressure-of-a-solution-that-was-prepared-at-91-f-by-dissolving-534-grams-of-aluminum-hydroxide-in-enough-water-to-generate-2-784-ml-of-solution",{"id":281,"category":28,"text_question":282,"slug":283},{"id":364,"category":28,"text_question":365,"slug":366},534299,"Three machines called A, B and C, produce 43%, 26% and 31% of the total production of\n a company, respectively. Furthermore, it has been detected that 8%, 2% and 1.6% of the product\n manufactured by these machines is defective.\n a) What is the probability that a product is not defective?\n b) A product is selected at random and found to be defective, what is the\n probability that it was manufactured on machine B?","three-machines-called-a-b-and-c-produce-43-26-and-31-of-the-total-production-of-a-company-respectively-furthermore-it-has-been-detected-that-8-2-and-1-6-of-the-product-manufactured-by-t",{"id":368,"category":28,"text_question":369,"slug":370},534437,"nI Exercises 65-68, the latitudes of a pair of cities are given. Assume that one city si directly south of the other and that the earth is a perfect sphere of radius 4000 miles. Use the arc length formula in terms of degrees to find the distance between the two cities.\n65. The North Pole: latitude 90° north Springfield, Illinois: latitude 40° north","ni-exercises-65-68-the-latitudes-of-a-pair-of-cities-are-given-assume-that-one-city-si-directly-south-of-the-other-and-that-the-earth-is-a-perfect-sphere-of-radius-4000-miles-use-the-arc-length-for",{"id":372,"category":28,"text_question":373,"slug":374},534440,"To find the increased amount on a standard term deposit with the following conditions:\n\nstarting amount: BGN 13000,\ntype of deposit: annual,\nannual compound interest rate: 1.4%,\nafter 4 years;","to-find-the-increased-amount-on-a-standard-term-deposit-with-the-following-conditions-starting-amount-bgn-13000-type-of-deposit-annual-annual-compound-interest-rate-1-4-after-4-years",{"id":376,"category":28,"text_question":377,"slug":378},534521,"Read the “Local Communities as Stakeholders: Does Amazon Really Need Tax Breaks?” example on p. 83 in Ch. 3 of Management: A Practical Introduction. In your response, discuss whether you feel that tax breaks for big companies benefit local communities. Describe ways to attract business to a region without having a negative impact on the larger community.","read-the-local-communities-as-stakeholders-does-amazon-really-need-tax-breaks-example-on-p-83-in-ch-3-of-management-a-practical-introduction-in-your-response-discuss-whether-you-feel-that-tax",{"id":380,"category":28,"text_question":381,"slug":382},534625,"Define excel and why we use it?","define-excel-and-why-we-use-it",{"id":384,"category":28,"text_question":385,"slug":386},534674,"Solve the system of equations by the addition method.\n\n0.01x-0.08y=-0.1\n0.2x+0.6y=0.2","solve-the-system-of-equations-by-the-addition-method-0-01x-0-08y-0-1-0-2x-0-6y-0-2",{"id":388,"category":28,"text_question":389,"slug":390},534689,"The car with an irresponsible driver starts to brake when it goes through a red light. When passing the traffic light, he does so at a speed of 115 kph in the right lane. Further ahead, 70 meters from the traffic light, a child is crossing the street and falls. If the effect of the car's brakes is equivalent to a deceleration of magnitude 5.7m/s². Is the child hit by the car or not? How far from the traffic light does the car stop?","the-car-with-an-irresponsible-driver-starts-to-brake-when-it-goes-through-a-red-light-when-passing-the-traffic-light-he-does-so-at-a-speed-of-115-kph-in-the-right-lane-further-ahead-70-meters-from",{"data":392},[393,397,401],{"id":394,"question":395,"answer":396},114848,"What is the length of the hypotenuse of a right triangle when the lengths of the other two sides are 4 and 7?","The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse length will be √(4^2 + 7^2) = √65, which is approximately 8.06 units.",{"id":398,"question":399,"answer":400},109544,"Math question: Given that log(base 2)5 = x and log(base 3)5 = y, what is the value of (x^2 + y^2) / (x + y)?","Answer: To find the value, we can use the logarithmic properties: x = log(base 2)5 implies 2^x = 5, and y = log(base 3)5 implies 3^y = 5. Substituting these into the expression, we get (2^x)^2 + (3^y)^2 / (2^x + 3^y). Simplifying further yields (25 + 25) / (5 + √15). Thus, the value is 50 / (5 + √15).",{"id":402,"question":403,"answer":404},148623,"What is the derivative of f(x) = 3x^4 - 5x^3 + 2x^2 - 6x + 1?","The derivative of f(x) is f'(x) = 12x^3 - 15x^2 + 4x - 6.",{"$sicons":406},{"bxl:facebook-circle":407,"bxl:instagram":411,"mdi:web":413,"la:apple":415,"ph:google-logo-bold":418,"ph:google-logo":421},{"left":408,"top":408,"width":409,"height":409,"rotate":408,"vFlip":177,"hFlip":177,"body":410},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":408,"top":408,"width":409,"height":409,"rotate":408,"vFlip":177,"hFlip":177,"body":412},"\u003Cpath fill=\"currentColor\" d=\"M11.999 7.377a4.623 4.623 0 1 0 0 9.248a4.623 4.623 0 0 0 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