Name: Solved: Determine the maximum number of points of intersection of 50 lines, knowing that 39 of them are concurrent at one point and the remaining ones are secant lines. - MathMaster
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:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> m = \\frac{9300 \\, \\text{g}}{1000} = 9.3 \\, \\text{kg} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Calculate the force due to gravity:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> F = m \\times g = 9.3 \\, \\text{kg} \\times 9.81 \\, \\text{m/s}^2 = 91.233 \\, \\text{N} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Convert the length of the hammer from centimeters to meters:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 18 \\, \\text{cm} = 0.18 \\, \\text{m} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Calculate the torque:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\tau = 0.18 \\, \\text{m} \\times 91.233 \\, \\text{N} = 16.42194 \\, \\text{Nm} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nThus, the torque about the pivot point due to the weight of the screw at the end of the hammer handle is approximately \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 16.42 \\, \\text{Nm} \u003C/math-field>\u003C/math-field>.",484,97,"suppose-you-have-to-use-a-hammer-in-your-hand-on-a-wall-with-a-screw-of-mass-9300-g-the-hammer-is-18-cm-long-calculate-the-torque-of-the-screw",{"id":51,"category":36,"text_question":52,"photo_question":38,"text_answer":53,"step_text_answer":8,"step_photo_answer":8,"views":54,"likes":55,"slug":56},538046," A cylinder makes six turns in 2 seconds, calculate: a) its angular velocity in rad/s; b) its period and c) its frequency.","a) Angular velocity

$\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\omega\u003C/math-field>\u003C/math-field>$ is given by the formula:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\omega = \\frac{\\Delta \\theta}{\\Delta t}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nHere, the cylinder makes 6 turns, and each turn is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2\\pi\u003C/math-field>\u003C/math-field> radians. Therefore, in 6 turns, the angle in radians is:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\Delta \\theta = 6 \\times 2\\pi = 12\\pi \\text{ radians}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nThe time period

$\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\Delta t\u003C/math-field>\u003C/math-field>$ is 2 seconds, so the angular velocity is:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\omega = \\frac{12\\pi}{2} = 6\\pi \\text{ rad/s}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nb) The period

$\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>T\u003C/math-field>\u003C/math-field>$ is the time it takes to complete one full rotation

$1 turn$ . Since the cylinder makes 6 turns in 2 seconds:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>T = \\frac{\\Delta t}{\\text{number of turns}} = \\frac{2}{6} = \\frac{1}{3} \\text{ s}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nc) The frequency

$\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>f\u003C/math-field>\u003C/math-field>$ is the reciprocal of the period:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>f = \\frac{1}{T} = \\frac{1}{\\frac{1}{3}} = 3 \\text{ Hz}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nThus, the answers are:\u003Cbr />\n\u003Cbr />\na) \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\omega = 6\\pi \\text{ rad/s}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nb) \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>T = \\frac{1}{3} \\text{ s}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nc) \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>f = 3 \\text{ Hz}\u003C/math-field>\u003C/math-field>",250,50,"a-cylinder-makes-six-turns-in-2-seconds-calculate-a-its-angular-velocity-in-rad-s-b-its-period-and-c-its-frequency",{"id":58,"category":36,"text_question":59,"photo_question":38,"text_answer":60,"step_text_answer":8,"step_photo_answer":8,"views":61,"likes":62,"slug":63},538045," A 73,000 g Ferris wheel accelerates from rest to an angular speed of 6.2 rad/s in 2 minutes. Considering the wheel as a hollow circular disk of radius 200 cm, calculate the net force on it?","1. Convert mass from grams to kilograms: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> m = 73,000 \\, \\text{g} = 73 \\, \\text{kg} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Convert radius from centimeters to meters:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> r = 200 \\, \\text{cm} = 2 \\, \\text{m} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Convert time from minutes to seconds: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> t = 2 \\, \\text{minutes} = 120 \\, \\text{seconds} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Calculate angular acceleration:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\alpha = \\frac{\\omega_f - \\omega_i}{t} = \\frac{6.2 \\, \\text{rad/s} - 0 \\, \\text{rad/s}}{120 \\, \\text{s}} = 0.0517 \\, \\text{rad/s}^2 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>5. Moment of inertia of a hollow circular disk:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> I = m \\cdot r^2 = 73 \\, \\text{kg} \\cdot

$2 \\, \\text{m}$ ^2 = 292 \\, \\text{kg} \\cdot \\text{m}^2 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>6. Calculate net torque:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\tau = I \\cdot \\alpha = 292 \\, \\text{kg} \\cdot \\text{m}^2 \\times 0.0517 \\, \\text{rad/s}^2 = 15.1044 \\, \\text{N} \\cdot \\text{m} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>7. Calculate net force

$since torque = force × radius$ :\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> F = \\frac{\\tau}{r} = \\frac{15.1044 \\, \\text{N} \\cdot \\text{m}}{2 \\, \\text{m}} = 7.5522 \\, \\text{N} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Rounding to a sensible number of significant figures gives the net force:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>F\\approx7.55\\,\\text{N}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Therefore, the net force on the Ferris wheel is approximately \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>7.55\\,\\text{N}\u003C/math-field>\u003C/math-field> .",655,131,"a-73-000-g-ferris-wheel-accelerates-from-rest-to-an-angular-speed-of-6-2-rad-s-in-2-minutes-considering-the-wheel-as-a-hollow-circular-disk-of-radius-200-cm-calculate-the-net-force-on-it",{"id":65,"category":36,"text_question":66,"photo_question":38,"text_answer":67,"step_text_answer":8,"step_photo_answer":8,"views":68,"likes":69,"slug":70},538044," A cylindrical piece of material 12 in in diameter is rotated on a lathe at 1300 rev/min. What is the tangential velocity at the surface of the cylinder? Value 4 points","1. Find angular velocity: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\omega = 1300 \\times 2\\pi \\text{ radians per minute} \u003C/math-field>\u003C/math-field>\u003Cbr />\n2. Calculate the radius: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> r = \\frac{12}{2} = 6 \\text{ inches} \u003C/math-field>\u003C/math-field>\u003Cbr />\n3. Use the formula for tangential velocity:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v = \\omega \\cdot r =

$1300 \\times 2\\pi$ \\cdot 6 = 15600\\pi \\text{ inches per minute} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nAnswer: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v = 15600\\pi \\text{ inches per minute} \u003C/math-field>\u003C/math-field>",649,130,"a-cylindrical-piece-of-material-12-in-in-diameter-is-rotated-on-a-lathe-at-1300-rev-min-what-is-the-tangential-velocity-at-the-surface-of-the-cylinder-value-4-points",{"id":72,"category":36,"text_question":73,"photo_question":38,"text_answer":74,"step_text_answer":8,"step_photo_answer":8,"views":75,"likes":76,"slug":77},538043," A pulley in a well with a diameter of 1200 cm initially rotates at 1.3 rev/s and then receives a constant angular acceleration of 3.12 rad/s2 . What is the tangential velocity of a belt mounted on said pulley? After 1 second, what is the tangential acceleration of the belt?","1. Calculate the initial angular velocity in radians per second: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\omega_0 = 1.3 \\, \\text{rev/s} \\times 2\\pi \\, \\text{rad/rev} = 2.6\\pi \\, \\text{rad/s} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Determine the final angular velocity after 1 second using the equation for angular velocity with constant acceleration:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\omega = \\omega_0 + \\alpha t \u003C/math-field>\u003C/math-field>\u003Cbr />\n where \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\alpha = 3.12 \\, \\text{rad/s}^2 \u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> t = 1 \\, \\text{s} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Calculate \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\omega \u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\omega = 2.6\\pi + 3.12 \\times 1 = 2.6\\pi + 3.12 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n4. Calculate the radius of the pulley:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> r = \\frac{1200 \\, \\text{cm}}{2} = 600 \\, \\text{cm} = 6 \\, \\text{m} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Find the tangential velocity \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v \u003C/math-field>\u003C/math-field> at \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> t = 1 \\, \\text{s} \u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v = \\omega \\times r \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v =

$2.6\\pi + 3.12$ \\times 6 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Simplify to find \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v \u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> v =

$8.168 + 3.12$ \\times 6 = 11.288 \\times 6 = 67.728 \\, \\text{m/s} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n7. Calculate the tangential acceleration \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> a_t \u003C/math-field>\u003C/math-field>, which is constant:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> a_t = \\alpha \\times r = 3.12 \\times 6 = 18.72 \\, \\text{m/s}^2 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n8. Final answers: \u003Cbr />\n Tangential velocity after 1 second: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 67.728 \\, \\text{m/s} \u003C/math-field>\u003C/math-field>\u003Cbr />\n Tangential acceleration: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 18.72 \\, \\text{m/s}^2 \u003C/math-field>\u003C/math-field>",955,191,"a-pulley-in-a-well-with-a-diameter-of-1200-cm-initially-rotates-at-1-3-rev-s-and-then-receives-a-constant-angular-acceleration-of-3-12-rad-s2-what-is-the-tangential-velocity-of-a-belt-mounted-on-s",{"id":79,"category":36,"text_question":80,"photo_question":38,"text_answer":81,"step_text_answer":8,"step_photo_answer":8,"views":82,"likes":83,"slug":84},538041,"Why do we use multiplication when dividing fractions. I know how to do the steps, but why do you keep the first number, changed to multiplication, and then write the reciprocal of the fraction.\n\nI guess I just don’t understand why we switched to multiplication. Thank you.","1. **Keep the First Fraction:** Write the first fraction as it is. \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{a}{b}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. **Change the Operation:** Change the division sign to a multiplication sign. \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\times\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. **Use the Reciprocal:** Use the reciprocal of the second fraction

$flip the numerator and denominator$ . \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{d}{c}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. **Multiply the Fractions:** Multiply the numerators and denominators. \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{a \\times d}{b \\times c}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. **Answer:** The division of the two given fractions as multiplication results in: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{a \\times d}{b \\times c}\u003C/math-field>\u003C/math-field>",702,140,"why-do-we-use-multiplication-when-dividing-fractions-i-know-how-to-do-the-steps-but-why-do-you-keep-the-first-number-changed-to-multiplication-and-then-write-the-reciprocal-of-the-fraction-i-gu",{"id":86,"category":36,"text_question":87,"photo_question":38,"text_answer":88,"step_text_answer":8,"step_photo_answer":8,"views":89,"likes":90,"slug":91},538040,"Prove the trig identity:\n\nSec^2x-Sin^2xSec^2x=1","1. Start with the left-hand side of the equation: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sec^2{x} - \\sin^2{x} \\sec^2{x} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Factor out \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sec^2{x} \u003C/math-field>\u003C/math-field>: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sec^2{x}

$1 - \\sin^2{x}$ \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Use the Pythagorean identity \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sin^2{x} + \\cos^2{x} = 1 \u003C/math-field>\u003C/math-field> to replace \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 1 - \\sin^2{x} \u003C/math-field>\u003C/math-field> with \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\cos^2{x} \u003C/math-field>\u003C/math-field>: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sec^2{x} \\cdot \\cos^2{x} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Substitute \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sec{x} = \\frac{1}{\\cos{x}} \u003C/math-field>\u003C/math-field>: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\left

$\\frac{1}{\\cos^2{x}}\\right$ \\cdot \\cos^2{x} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Simplify: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 1 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nTherefore, the identity is proven: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sec^2{x} - \\sin^2{x} \\sec^2{x} = 1 \u003C/math-field>\u003C/math-field>",294,59,"prove-the-trig-identity-sec-2x-sin-2xsec-2x-1",{"id":93,"category":36,"text_question":94,"photo_question":38,"text_answer":95,"step_text_answer":8,"step_photo_answer":8,"views":96,"likes":97,"slug":98},538039,"Prove the trig identity:\n\n1+Sec^2x/Sec^2x = 1 + cos^2x","1. Start with the left-hand side of the equation: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1+\\sec^2 x}{\\sec^2 x}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Recall that \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sec x = \\frac{1}{\\cos x}\u003C/math-field>\u003C/math-field> , so \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sec^2 x = \\frac{1}{\\cos^2 x}\u003C/math-field>\u003C/math-field> . Therefore, substitute:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1+\\frac{1}{\\cos^2 x}}{\\frac{1}{\\cos^2 x}}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Simplify the fraction:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{\\cos^2x+1}{\\frac{\\cos^2\\left

$x\\right$ }{\\cos^2x}}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Multiply by the reciprocal of the denominator:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>

$\\cos^2x+1$ \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>6. Simplify to get:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\cos^2 x + 1\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>7. Compare with the right-hand side, which is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1 + \\cos^2 x\u003C/math-field>\u003C/math-field> . Since \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\cos^2 x + 1\u003C/math-field>\u003C/math-field> is equivalent to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1 + \\cos^2 x\u003C/math-field>\u003C/math-field> , the trigonometric identity is proved.\u003Cbr>\u003Cbr>Answer: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1 + \\cos^2 x\u003C/math-field>\u003C/math-field>",1031,206,"prove-the-trig-identity-1-sec-2x-sec-2x-1-cos-2x",{"id":100,"category":36,"text_question":101,"photo_question":38,"text_answer":102,"step_text_answer":8,"step_photo_answer":8,"views":103,"likes":104,"slug":105},538038,"Prove the trig identity:\n\nCos x/1-sinx - Cos x/1+sin x = 2tan x","\u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba

$0, 0, 0, .3$ ;box-shadow: 0 0 0 rgba

$0, 0, 0, .2$ \n\" read-only>

$=\\frac{\\sin(2x)}{(-\\sin(x)+1)(\\sin(x)+1)}$ \u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba

$0, 0, 0, .3$ ;box-shadow: 0 0 0 rgba

$0, 0, 0, .2$ \n\" read-only>

$=\\frac{\\sin(2x)}{\\cos^{2}(x)}$ \u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba

$0, 0, 0, .3$ ;box-shadow: 0 0 0 rgba

$0, 0, 0, .2$ \n\" read-only>

$=2\\tan(x)$ \u003C/math-field>\n \u003Cbr>\n \u003C/div>",770,154,"prove-the-trig-identity-cos-x-1-sinx-cos-x-1-sin-x-2tan-x",{"id":107,"category":36,"text_question":108,"photo_question":38,"text_answer":109,"step_text_answer":8,"step_photo_answer":8,"views":110,"likes":111,"slug":112},538037,"Prove the trig identity:\n\n1+cos x/sin x = csc x + cot x","1. Start with the right-hand side of the equation: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\csc x + \\cot x \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cbr />\n2. Express \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\csc x \u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\cot x \u003C/math-field>\u003C/math-field> in terms of sine and cosine:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\csc x = \\frac{1}{\\sin x} \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\cot x = \\frac{\\cos x}{\\sin x} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Add the fractions:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\csc x + \\cot x = \\frac{1}{\\sin x} + \\frac{\\cos x}{\\sin x} = \\frac{1 + \\cos x}{\\sin x} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. The expression \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{1 + \\cos x}{\\sin x} \u003C/math-field>\u003C/math-field> matches the left-hand side of the identity.\u003Cbr />\n\u003Cbr />\nTherefore, the identity is valid: \u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{1 + \\cos x}{\\sin x} = \\csc x + \\cot x \u003C/math-field>\u003C/math-field>",605,121,"prove-the-trig-identity-1-cos-x-sin-x-csc-x-cot-x",{"id":114,"category":36,"text_question":115,"photo_question":38,"text_answer":116,"step_text_answer":8,"step_photo_answer":8,"views":117,"likes":118,"slug":119},538036,"Prove the trig identity:\n\n

$1-cos θ$

$1+cos θ$ = 1/csc^2 θ","1. Start with the left side of the equation: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>

$1 - \\cos \\theta$

$1 + \\cos \\theta$ \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Apply the difference of squares formula: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>

$1 - \\cos \\theta$

$1 + \\cos \\theta$ = 1 - \\cos^2 \\theta\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Use the Pythagorean identity: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1 - \\cos^2 \\theta = \\sin^2 \\theta\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Relate to the right side: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sin^2 \\theta = \\frac{1}{\\csc^2 \\theta}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>",1096,219,"prove-the-trig-identity-1-cos-1-cos-1-csc-2",{"id":121,"category":36,"text_question":122,"photo_question":38,"text_answer":123,"step_text_answer":8,"step_photo_answer":8,"views":124,"likes":125,"slug":126},538035,"Prove the trig identity:\n\n Sec θ - Cos θ/Sec θ = Sin^2 θ","1. Start with the left-hand side

$LHS$ : \u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{LHS} = \\frac{\\sec \\theta - \\cos \\theta}{\\sec \\theta}\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n2. Rewrite the terms in terms of sine and cosine:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sec \\theta = \\frac{1}{\\cos \\theta}\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n3. Substitute into the LHS:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{LHS} = \\frac{\\frac{1}{\\cos \\theta} - \\cos \\theta}{\\frac{1}{\\cos \\theta}}\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n4. Simplify the expression inside the fraction:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= \\frac{\\frac{1 - \\cos^2 \\theta}{\\cos \\theta}}{\\frac{1}{\\cos \\theta}}\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n5. Simplify further by multiplying by the reciprocal:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>=

$1 - \\cos^2 \\theta$ \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Substitute the Pythagorean identity:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= \\sin^2 \\theta\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\nHence, the identity is proven since:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{\\sec \\theta - \\cos \\theta}{\\sec \\theta} = \\sin^2 \\theta\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\nSo, the right-hand side

$RHS$ and transformed LHS are equal. \u003Cbr />\n\u003Cbr />\nThis completes the proof of the identity.",1434,287,"prove-the-trig-identity-sec-cos-sec-sin-2",{"id":128,"category":36,"text_question":129,"photo_question":38,"text_answer":130,"step_text_answer":8,"step_photo_answer":8,"views":131,"likes":132,"slug":133},538032,"How much is 18% of a kilometer?","18% of a kilometer can be calculated by multiplying 18% with the length of a kilometer. \u003Cbr />\n\u003Cbr />\n[SOLUTION] \u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 180 \\text{ meters} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n[STEP-BY-STEP]\u003Cbr />\n\u003Cbr />\n1. Convert percentage to a decimal.\u003Cbr />\n - 18% is the same as 0.18.\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 18\\% = \\frac{18}{100} = 0.18 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. A kilometer is 1,000 meters.\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 1 \\text{ kilometer} = 1000 \\text{ meters} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Calculate 18% of a kilometer in meters.\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 0.18 \\times 1000 = 180 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Therefore, 18% of a kilometer is:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 180 \\text{ meters} \u003C/math-field>\u003C/math-field>",520,104,"how-much-is-18-of-a-kilometer",{"id":135,"category":36,"text_question":136,"photo_question":38,"text_answer":137,"step_text_answer":8,"step_photo_answer":8,"views":138,"likes":139,"slug":140},538031,"How much is 10% of a kilogram?","1. A kilogram is equal to 1000 grams. \u003Cbr />\n\u003Cbr />\n2. To find 10% of a kilogram, calculate 10% of 1000 grams: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.1 \\times 1000 = 100\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n3. Convert the result back to kilograms:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>100 \\, \\text{grams} = 0.1 \\, \\text{kg}\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\nThus, 10% of a kilogram is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.1 \\, \\text{kg}\u003C/math-field>\u003C/math-field>.",763,153,"how-much-is-10-of-a-kilogram",{"id":142,"category":36,"text_question":143,"photo_question":38,"text_answer":144,"step_text_answer":8,"step_photo_answer":8,"views":145,"likes":146,"slug":147},538030,"98/100 \nWrite the fraction as a decimal","1. Begin with the fraction \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{98}{100}\u003C/math-field>\u003C/math-field>. \u003Cbr />\n\u003Cbr />\n2. Convert the fraction to a decimal by dividing the numerator by the denominator: \u003Cbr />\n \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{98}{100} = 98 \\div 100 = 0.98\u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cbr />\n3. Thus, the fraction \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{98}{100}\u003C/math-field>\u003C/math-field> is equivalent to the decimal 0.98.\u003Cbr />\n\u003Cbr />\nAnswer: 0.98",383,77,"98-100-write-the-fraction-as-a-decimal",{"id":149,"category":36,"text_question":150,"photo_question":38,"text_answer":151,"step_text_answer":8,"step_photo_answer":8,"views":152,"likes":153,"slug":154},538029,"

$-3,-2$ and

$-8, -2$ \nFind the distance between 2 points","To find the distance between two points \$(-3,-2)\$ and \$(-8, -2)\$ in a 2D coordinate plane, we use the distance formula:\u003Cbr>\u003Cbr>1. The distance formula is: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Substitute the given points \$(-3, -2)\$ and \$(-8, -2)\$ into the formula:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> d = \\sqrt{((-8) - (-3))^2 + ((-2) - (-2))^2} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Simplify the expression inside the square root:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> d = \\sqrt{(-8 + 3)^2 + (0)^2} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> d = \\sqrt{(-5)^2 + 0} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Calculate:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> d = \\sqrt{25} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>5. Hence, the distance between the points is:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5\u003C/math-field>\u003C/math-field> units",1078,216,"3-2-and-8-2-find-the-distance-between-2-points",{"id":156,"category":36,"text_question":157,"photo_question":38,"text_answer":158,"step_text_answer":8,"step_photo_answer":8,"views":159,"likes":160,"slug":161},538028,"2.01c-6=-0.15c+6.96","Solution:\u003Cbr />\n1. Start with the given equation:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2.01c - 6 = -0.15c + 6.96\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Move all terms involving \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c\u003C/math-field>\u003C/math-field> to one side by adding \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.15c\u003C/math-field>\u003C/math-field> to both sides:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2.01c + 0.15c = 6.96 + 6\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Simplify the equation:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2.16c = 12.96\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c\u003C/math-field>\u003C/math-field> by dividing both sides by \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2.16\u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c = \\frac{12.96}{2.16}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Calculate the value:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>c = 6\u003C/math-field>\u003C/math-field>",865,173,"2-01c-6-0-15c-6-96",{"id":163,"category":36,"text_question":164,"photo_question":38,"text_answer":165,"step_text_answer":8,"step_photo_answer":8,"views":166,"likes":167,"slug":168},538027,"1/4x+13>0.25(2x-32)","Solution:\u003Cbr />\n1. Given inequality:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{4}x + 13 > 0.25(2x - 32)\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Simplify the right side:\u003Cbr />\n * Convert 0.25 to a fraction: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.25 = \\frac{1}{4}\u003C/math-field>\u003C/math-field>\u003Cbr />\n * Distribute: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{4}(2x - 32) = \\frac{1}{4} \\cdot 2x - \\frac{1}{4} \\cdot 32 = \\frac{1}{2}x - 8\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Rewrite the inequality:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{4}x + 13 > \\frac{1}{2}x - 8\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Eliminate fractions by multiplying all terms by 4:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x + 52 > 2x - 32\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Rearrange terms:\u003Cbr />\n * Subtract \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> from both sides:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>52 > x - 32\u003C/math-field>\u003C/math-field>\u003Cbr />\n * Add 32 to both sides:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>84 > x\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. The solution to the inequality is:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x \u003C 84\u003C/math-field>\u003C/math-field>",301,60,"1-4x-13-0-25-2x-32",{"id":170,"category":36,"text_question":171,"photo_question":38,"text_answer":172,"step_text_answer":8,"step_photo_answer":8,"views":173,"likes":174,"slug":175},538026,"0.5(4x+24)=22x-2","Solution:\u003Cbr />\n1. Given equation:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.5(4x + 24) = 22x - 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Simplify the left side:\u003Cbr />\n- Apply the distributive property: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.5 \\times 4x + 0.5 \\times 24 = 22x - 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n- This gives us: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2x + 12 = 22x - 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Rearrange to solve for x:\u003Cbr />\n- Subtract \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2x\u003C/math-field>\u003C/math-field> from both sides: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2x + 12 - 2x = 22x - 2 - 2x\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Simplify: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>12 = 20x - 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Add 2 to both sides to isolate terms:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>12 + 2 = 20x - 2 + 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Simplify: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>14 = 20x\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Divide by 20 to solve for x:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{14}{20}\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Simplify the fraction: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{7}{10}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nThe solution to the equation is $x = \\\\frac{7}{10}$.",614,123,"0-5-4x-24-22x-2",{"first":6,"last":177,"prev":8,"next":10},186,{"current_page":6,"from":6,"last_page":177,"links":179,"path":213,"per_page":214,"to":214,"total":215},[180,183,186,188,190,192,194,197,200,203,206,209,211],{"url":6,"label":181,"active":182},"1",true,{"url":10,"label":184,"active":185},"2",false,{"url":13,"label":187,"active":185},"3",{"url":16,"label":189,"active":185},"4",{"url":19,"label":191,"active":185},"5",{"url":22,"label":193,"active":185},"6",{"url":195,"label":196,"active":185},7,"7",{"url":198,"label":199,"active":185},8,"8",{"url":201,"label":202,"active":185},9,"9",{"url":204,"label":205,"active":185},10,"10",{"url":207,"label":208,"active":185},185,"185",{"url":177,"label":210,"active":185},"186",{"url":10,"label":212,"active":185},"Next »","https://api.math-master.org/api/question",20,3704,{"data":217},{"questions":218},[219,223,227,230,234,238,242,246,250,254,258,262,266,270,274,278,282,286,290,294],{"id":220,"category":36,"text_question":221,"slug":222},532009,"How much volume of water in MegaLiters (ML) is required to irrigate 30 Hectare crop\narea with depth of 20mm?","how-much-volume-of-water-in-megaliters-ml-is-required-to-irrigate-30-hectare-crop-area-with-depth-of-20mm",{"id":224,"category":36,"text_question":225,"slug":226},532307,"Write 32/25 as a percent","write-32-25-as-a-percent",{"id":228,"category":36,"text_question":229,"slug":229},533938,"132133333-33",{"id":231,"category":36,"text_question":232,"slug":233},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":235,"category":36,"text_question":236,"slug":237},534015,"(5u + 6)-(3u+2)=","5u-6-3u-2",{"id":239,"category":36,"text_question":240,"slug":241},534043,"The equation of the circle that passes through (5,3) and is tangent to the abscissa axis at x=2 is\n a.(x-2)^2 (y 3)^2 = 9\n b.(x-2)^2 (y-3)^2 = 9\n c.(x-2)^2 (y-3)^2 = 4\n d.(x-2)^2 (y 1)^2 = 4\n e.(x-2)^2 (y-1)^2 = 4","the-equation-of-the-circle-that-passes-through-5-3-and-is-tangent-to-the-abscissa-axis-at-x-2-is-a-x-2-2-y-3-2-9-b-x-2-2-y-3-2-9-c-x-2-2-y-3-2-4-d-x-2-2-y-1-2-4-e-x-2",{"id":243,"category":36,"text_question":244,"slug":245},534078,"is the x element (180,270), if tanx-3cotx=2, sinx ?","is-the-x-element-180-270-if-tanx-3cotx-2-sinx",{"id":247,"category":36,"text_question":248,"slug":249},534134,"find x in the equation\n 2x-4=6","find-x-in-the-equation-2x-4-6",{"id":251,"category":36,"text_question":252,"slug":253},534235,"Equine infectious anemia (EIA) is considered the main infectious disease in Brazilian equine farming, for which there is no effective vaccine or treatment. It is caused by a retrovirus of the genus Lentivirus, which affects horses, donkeys and mules and is transmitted in nature mainly by hematophagous insects of the genus Tabanidae.\n\n Researchers analyzed the records of 9,439 equids from Acre, submitted to the agar gel immunodiffusion test (AGID) for equine infectious anemia (EIA), between 1986 and 1996. Of these, 6199 tested positive for equine infectious anemia (EIA) .\n\n Knowing that the age of AIE-positive horses follows a Normal distribution with a mean of 5 years and a standard deviation of 1.5 years, determine the expected number of AIE-positive horses in the Acre sample that will be aged less than or equal to 3 years.\n\n\n\n ATTENTION: Provide the answer to exactly FOUR decimal places.","equine-infectious-anemia-eia-is-considered-the-main-infectious-disease-in-brazilian-equine-farming-for-which-there-is-no-effective-vaccine-or-treatment-it-is-caused-by-a-retrovirus-of-the-genus-le",{"id":255,"category":36,"text_question":256,"slug":257},534277,"In an audience of 4000 people, 2 people are chosen, at random, to appear on stage. How many ways can the people be chosen?","in-an-audience-of-4000-people-2-people-are-chosen-at-random-to-appear-on-stage-how-many-ways-can-the-people-be-chosen",{"id":259,"category":36,"text_question":260,"slug":261},534309,"Determine the increase of the function y=4x−5 when the argument changes from x1=2 to x2=3","determine-the-increase-of-the-function-y-4x-5-when-the-argument-changes-from-x1-2-to-x2-3",{"id":263,"category":36,"text_question":264,"slug":265},534394,"In a physics degree course, there is an average dropout of 17 students in the\n first semester. What is the probability that the number of dropouts in the first\n semester in a randomly selected year has between 13 and 16 students?","in-a-physics-degree-course-there-is-an-average-dropout-of-17-students-in-the-first-semester-what-is-the-probability-that-the-number-of-dropouts-in-the-first-semester-in-a-randomly-selected-year-ha",{"id":267,"category":36,"text_question":268,"slug":269},534408,"Which statement best describes the key changes in perspectives on inclusion?\r\n\r\n\t\t\r\nAn inclusive program must consider the unique experiences of every child and family as well as the child's strengths and needs.\r\n\r\n\t\t\r\nThere is a shift in thinking about individual programs as \"inclusive programs\" to thinking about inclusion as something that reflects the cultural influence of the family.\r\n\r\n\t\t\r\nThere is a greater emphasis on barriers to full participation and the acknowledgement that all children are unique and must be fully and meaningfully engaged in a program.\r\n\r\n\t\t\r\nIn an inclusive program all participants are accepted by their peers and other members of the community.","which-statement-best-describes-the-key-changes-in-perspectives-on-inclusion-an-inclusive-program-must-consider-the-unique-experiences-of-every-child-and-family-as-well-as-the-child-s-strengths",{"id":271,"category":36,"text_question":272,"slug":273},534432,"Calculate the change in internal energy of a gas that receives 16000 J of heat at constant pressure (1.3 atm) expanding from 0.100 m3 to 0.200 m3.\n\n Question 1Answer\n to.\n 7050J\n\n b.\n 2125J\n\n c.\n None of the above\n\n d.\n 2828J\n\n and.\n 10295 J","calculate-the-change-in-internal-energy-of-a-gas-that-receives-16000-j-of-heat-at-constant-pressure-1-3-atm-expanding-from-0-100-m3-to-0-200-m3-question-1answer-to-7050j-b-2125j-c-none",{"id":275,"category":36,"text_question":276,"slug":277},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":279,"category":36,"text_question":280,"slug":281},534524,"x²-7x+12=0","x-7x-12-0",{"id":283,"category":36,"text_question":284,"slug":285},534533,"8/9 divided by 10/6","8-9-divided-by-10-6",{"id":287,"category":36,"text_question":288,"slug":289},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":291,"category":36,"text_question":292,"slug":293},534638,"Sin(5pi/3)","sin-5pi-3",{"id":295,"category":36,"text_question":296,"slug":297},534664,"15=5(x+3)","15-5-x-3",{"data":299},{"id":300,"category":36,"slug":301,"text_question":302,"photo_question":8,"text_answer":303,"step_text_answer":8,"step_photo_answer":8,"views":304,"likes":305,"expert":306},537675,"determine-the-maximum-number-of-points-of-intersection-of-50-lines-knowing-that-39-of-them-are-concurrent-at-one-point-and-the-remaining-ones-are-secant-lines","Determine the maximum number of points of intersection of 50 lines, knowing that 39 of them are concurrent at one point and the remaining ones are secant lines.","1. Calcular el número de intersecciones para 11 rectas no concurrentes: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\binom{11}{2} = 55\u003C/math-field>\u003C/math-field>.\u003Cbr />\n2. Las 39 rectas concurrentes se cruzan en un solo punto: suman 1 intersección total.\u003Cbr />\n3. Sumar las intersecciones identificadas: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1 + 55 = 56\u003C/math-field>\u003C/math-field>.\u003Cbr />\n4. Respuesta final: 56.",910,182,{"id":22,"name":307,"photo":308,"biography":309,"created_at":8,"updated_at":8,"rating":310,"total_answer":311},"Jett","https://api.math-master.org/img/experts/6/6.webp","Hello,\nI am pleased to introduce myself as a Mathmaster Expert 😊\nI am Engr. Paul Eejay Dy, a Registered Civil Engineer in the Philippines and Mathematics has always been my favorite subject.\nStudying Mathematics is a really difficult but fun experience for me and with the help of really great teachers and professors that helped me on this journey, I overcame all the obstacles that came my way until I achieved my dream profession.\nSo for all Mathmaster users, it is a great pleasure to be part of your journey into this rough but fulfilling journey of studying Mathematics. Study hard, and keep fighting for your dreams 😇🙏",4.7,94,{"data":313},{"questions":314},[315,319,323,327,331,335,339,343,347,351,355,359,363,367,371,375,379,383,387,391],{"id":316,"category":36,"text_question":317,"slug":318},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":320,"category":36,"text_question":321,"slug":322},533954,"3x+2/2x-1 + 3+x/2x-1 - 3x-2/2x-1","3x-2-2x-1-3-x-2x-1-3x-2-2x-1",{"id":324,"category":36,"text_question":325,"slug":326},533961,"2.5 / 21.85","2-5-21-85",{"id":328,"category":36,"text_question":329,"slug":330},534026,"\"If three wolves catch three rabbits in three hours, how many wolves would it take to catch a hundred rabbits in a hundred hours?\"\n\n The answer is the number of response units.","if-three-wolves-catch-three-rabbits-in-three-hours-how-many-wolves-would-it-take-to-catch-a-hundred-rabbits-in-a-hundred-hours-the-answer-is-the-number-of-response-units",{"id":332,"category":36,"text_question":333,"slug":334},534048,"To celebrate the five-year anniversary of a consultancy specializing in information technology, the administrator decided to draw 3 different qualification courses among its 10 employees. Considering that the same employee cannot be drawn more than once, the total number of different ways of drawing among employees is:","to-celebrate-the-five-year-anniversary-of-a-consultancy-specializing-in-information-technology-the-administrator-decided-to-draw-3-different-qualification-courses-among-its-10-employees-considering",{"id":336,"category":36,"text_question":337,"slug":338},534051,"2.3/-71.32","2-3-71-32",{"id":340,"category":36,"text_question":341,"slug":342},534055,"7/6-(-1/9)","7-6-1-9",{"id":344,"category":36,"text_question":345,"slug":346},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":348,"category":36,"text_question":349,"slug":350},534061,"In a store, a person carries 14 kilos of rice and 28 kilos of flour. In what ratio are the kilos found? (Remember to simplify until you reach an irreducible fraction)","in-a-store-a-person-carries-14-kilos-of-rice-and-28-kilos-of-flour-in-what-ratio-are-the-kilos-found-remember-to-simplify-until-you-reach-an-irreducible-fraction",{"id":352,"category":36,"text_question":353,"slug":354},534125,"There were no defectives in a sample of 1 light bulb does this sample provide sufficient evidence that in the warehouse with millions of light bulbs fewer than 10% are defective?","there-were-no-defectives-in-a-sample-of-1-light-bulb-does-this-sample-provide-sufficient-evidence-that-in-the-warehouse-with-millions-of-light-bulbs-fewer-than-10-are-defective",{"id":356,"category":36,"text_question":357,"slug":358},534139,"(24, -7) is on the terminal arm of an angle in standard position.\nDetermine the exact values of the primary trigonometric functions.","24-7-is-on-the-terminal-arm-of-an-angle-in-standard-position-determine-the-exact-values-of-the-primary-trigonometric-functions",{"id":360,"category":36,"text_question":361,"slug":362},534219,"The Humane Society has asked for our help again this week. Currently they are charging $50 for an adoption fee. Unfortunately they just pulled this number out of the air and do not know why they are charging this amount. They would like to charge an amount that covers all the adoption costs – both the variable costs for adoptions as well as the fixed cost for the kennel portion of the Humane Shelter operations. We can help them by doing a breakeven analysis.\r\n\r\nDuring a client meeting we gathered these facts. There are 2 part-time employees that each earn $1000 per month. The utilities for the kennel area (water, electricity) are $200 per month. The average food cost for animals in the kennel is $800 per month. In addition, each animal that is adopted receives a rabies vaccination that costs $4 and is micro-chipped that costs $6. \r\n\r\nAt the current cost of $50, how many animals must be adopted to break-even?\r\nWhat would break-even be at a $60 adoption fee? \r\nWhat would break-even be if the fee were lowered to $40?\r\nThe newspaper has suggested that the Humane Society advertise to increase pet adoptions. The package that they have recommended costs $1000 for a very small ad run every day for a month. If the Humane Society does this extra advertising, how will it affect breakeven?\r\nBased on what you have learned about elasticity, what price do you recommend for the adoption fee?","the-humane-society-has-asked-for-our-help-again-this-week-currently-they-are-charging-50-for-an-adoption-fee-unfortunately-they-just-pulled-this-number-out-of-the-air-and-do-not-know-why-they-are-c",{"id":364,"category":36,"text_question":365,"slug":366},534460,"(6²-14)÷11•(-3)","6-14-11-3",{"id":368,"category":36,"text_question":369,"slug":370},534470,"A company dedicated to the manufacture of shirts sells the units at a price of $40, the cost of each shirt is $24, a commission is paid for the sale of a unit of shirt of $2 and its fixed costs are $3500\n Determine the marginal contribution","a-company-dedicated-to-the-manufacture-of-shirts-sells-the-units-at-a-price-of-40-the-cost-of-each-shirt-is-24-a-commission-is-paid-for-the-sale-of-a-unit-of-shirt-of-2-and-its-fixed-costs-are-3",{"id":372,"category":36,"text_question":373,"slug":374},534481,"2x-5-x+2=5x-11","2x-5-x-2-5x-11",{"id":376,"category":36,"text_question":377,"slug":378},534484,"00 piece jigsaw puzzle. the completed puzzle is 10x10. each piech connects to at least 2 other pieces. i plan to assemble by taking pieces out of box one by one. if i've already taken out 2 pieces that dont directly connect, what is the minimum number of additional pieces that i need to draw to in order to guarentee that the original 2 pieces connect?","00-piece-jigsaw-puzzle-the-completed-puzzle-is-10x10-each-piech-connects-to-at-least-2-other-pieces-i-plan-to-assemble-by-taking-pieces-out-of-box-one-by-one-if-i-ve-already-taken-out-2-pieces-tha",{"id":380,"category":36,"text_question":381,"slug":382},534560,"Identify the slope and y intercept \ny=11+2/3x","identify-the-slope-and-y-intercept-y-11-2-3x",{"id":384,"category":36,"text_question":385,"slug":386},534622,"It costs a manufacturer $2,500 to purchase the tools to manufacture a certain homemade item. If the cost for materials and labor is 60¢ per item produced, and if the manufacturer can sell each item for 90¢, find how many items must he produce and sell to make a profit of $2000?","it-costs-a-manufacturer-2-500-to-purchase-the-tools-to-manufacture-a-certain-homemade-item-if-the-cost-for-materials-and-labor-is-60-per-item-produced-and-if-the-manufacturer-can-sell-each-item-fo",{"id":388,"category":36,"text_question":389,"slug":390},534684,"The domain of the function f(x)=x+7x2−144 \nis (−∞,), ( ,), and ( , ∞).","the-domain-of-the-function-f-x-x-7x2-144-is-and",{"id":392,"category":36,"text_question":393,"slug":394},534686,"To apply a diagnostic test, in how many ways can 14 students be chosen out of 25? if the order does not matter","to-apply-a-diagnostic-test-in-how-many-ways-can-14-students-be-chosen-out-of-25-if-the-order-does-not-matter",{"data":396},[397,401,405],{"id":398,"question":399,"answer":400},102881,"What is 3/5 as a percent?","To convert 3/5 to a percent, we can multiply it by 100. So, 3/5 is equal to 60%.",{"id":402,"question":403,"answer":404},139288,"What is the domain of the trigonometric function f(x) = sin(x)?","The domain of f(x) = sin(x) is all real numbers, or (-∞, ∞). Since the sine function repeats itself indefinitely, it can take any value for any real number input.",{"id":406,"question":407,"answer":408},108977,"Question: What is the value of x when log(x) = 2?","Answer: The value of x can be obtained by exponentiating both sides of the equation, giving x = 10² = 100.",{"$sicons":410},{"bxl:facebook-circle":411,"bxl:instagram":415,"mdi:web":417,"la:apple":419,"ph:google-logo-bold":422,"ph:google-logo":425},{"left":412,"top":412,"width":413,"height":413,"rotate":412,"vFlip":185,"hFlip":185,"body":414},0,24,"\u003Cpath fill=\"currentColor\" d=\"M12.001 2.002c-5.522 0-9.999 4.477-9.999 9.999c0 4.99 3.656 9.126 8.437 9.879v-6.988h-2.54v-2.891h2.54V9.798c0-2.508 1.493-3.891 3.776-3.891c1.094 0 2.24.195 2.24.195v2.459h-1.264c-1.24 0-1.628.772-1.628 1.563v1.875h2.771l-.443 2.891h-2.328v6.988C18.344 21.129 22 16.992 22 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