Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

A 39-meter ladder is sliding down a vertical wall so the distance between the bottom of the ladder and the wall is increasing at 10 meters per minute. At a certain instant, the bottom of the ladder is 36 meters from the wall. What is the rate of change of the distance between the top of the ladder and the ground at that instant (in meters per minute)? Choose 1 answer: (A) -12 (B) -(25)/(6) (C) -24 (D) -10

A 3939-meter ladder is sliding down a vertical wall so the distance between the bottom of the ladder and the wall is increasing at 1010 meters per minute.\newlineAt a certain instant, the bottom of the ladder is 36 \mathbf{3 6} meters from the wall.\newlineWhat is the rate of change of the distance between the top of the ladder and the ground at that instant (in meters per minute)?\newlineChoose 11 answer:\newline(A) −12-12\newline(B) −256 -\frac{25}{6} \newline(C) −24-24\newline(D) −10-10

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Grade 7right chevron icon
Calculate unit rates with fractionsright chevron icon

Full solution

Q. A 3939-meter ladder is sliding down a vertical wall so the distance between the bottom of the ladder and the wall is increasing at 1010 meters per minute.\newlineAt a certain instant, the bottom of the ladder is 36 \mathbf{3 6} meters from the wall.\newlineWhat is the rate of change of the distance between the top of the ladder and the ground at that instant (in meters per minute)?\newlineChoose 11 answer:\newline(A) −12-12\newline(B) −256 -\frac{25}{6} \newline(C) −24-24\newline(D) −10-10
  1. Triangle Description: We're dealing with a right triangle where the ladder is the hypotenuse, the distance from the wall is one leg, and the height of the ladder above the ground is the other leg.
  2. Variable Assignment: Let's call the distance from the wall to the bottom of the ladder xx, and the height of the ladder above the ground yy. We know that xx is changing at 1010 meters per minute.
  3. Pythagoras' Theorem: Using Pythagoras' theorem, we have x2+y2=392x^2 + y^2 = 39^2, since the ladder is 3939 meters long.
  4. Differentiation: Differentiate both sides of the equation with respect to time tt to find the rate of change of yy with respect to time. So we get 2x(dxdt)+2y(dydt)=02x\left(\frac{dx}{dt}\right) + 2y\left(\frac{dy}{dt}\right) = 0.
  5. Initial Values: We know dxdt=10\frac{dx}{dt} = 10 meters per minute (since xx is increasing at 1010 meters per minute), and at the instant we're considering, x=36x = 36 meters.
  6. Calculate yy: We need to find yy at the instant x=36x = 36 meters. Using Pythagoras' theorem again, we have 362+y2=39236^2 + y^2 = 39^2.
  7. Substitute Values: Calculate y2=392−362=1521−1296=225y^2 = 39^2 - 36^2 = 1521 - 1296 = 225.
  8. Solve for dydt\frac{dy}{dt}: Take the square root of y2y^2 to find yy. So y=225=15y = \sqrt{225} = 15 meters.
  9. Isolate dydt\frac{dy}{dt}: Now we can substitute xx, dxdt\frac{dx}{dt}, and yy into the differentiated equation: 2⋅36⋅10+2⋅15⋅(dydt)=02\cdot36\cdot10 + 2\cdot15\cdot\left(\frac{dy}{dt}\right) = 0.
  10. Final Result: Solve for dydt\frac{dy}{dt}: 720+30⋅dydt=0720 + 30\cdot\frac{dy}{dt} = 0.
  11. Final Result: Solve for dydt\frac{dy}{dt}: 720+30(dydt)=0720 + 30\left(\frac{dy}{dt}\right) = 0.Isolate dydt\frac{dy}{dt}: 30(dydt)=−72030\left(\frac{dy}{dt}\right) = -720.
  12. Final Result: Solve for dydt\frac{dy}{dt}: 720+30(dydt)=0720 + 30\left(\frac{dy}{dt}\right) = 0.Isolate dydt\frac{dy}{dt}: 30(dydt)=−72030\left(\frac{dy}{dt}\right) = -720.Divide both sides by 3030 to find dydt\frac{dy}{dt}: dydt=−72030=−24\frac{dy}{dt} = \frac{-720}{30} = -24 meters per minute.

More problems from Calculate unit rates with fractions

Question
What is the period of\newliney=cos⁡(−4πx+3)−7? y=\cos (-4 \pi x+3)-7 ? \newlineGive an exact value.\newlineunits
Get tutor helpright-arrow

Posted 9 months ago

Question
Mandy works construction. She knows that a 55 meter long metal bar has a mass of 40 kg 40 \mathrm{~kg} . Mandy wants to figure out the mass (w) (w) of a bar made out of the same metal that is 33 meters long and the same thickness.\newlineWhat is the mass of the shorter bar?\newlinekg \mathrm{kg}
Get tutor helpright-arrow

Posted 9 months ago

Question
Mika can eat 2121 hot dogs in 66 minutes. She wants to know how many minutes (m) (m) it would take her to eat 3535 hot dogs if she can keep up the same pace.\newlineHow many minutes would Mika need to eat 3535 hot dogs?\newlineminutes
Get tutor helpright-arrow

Posted 9 months ago

Question
Elton is a candle maker. Each 15 cm 15 \mathrm{~cm} long candle he makes burns evenly for 66 hours.\newlineIf Elton makes a 45 cm 45 \mathrm{~cm} long candle, how long would it burn?\newlinehours
Get tutor helpright-arrow

Posted 9 months ago

Question
Harvey the wonder hamster can run 316 km 3 \frac{1}{6} \mathrm{~km} in 14 \frac{1}{4} hour. Harvey runs at a constant rate.\newlineFind his average speed in kilometers per hour.\newlinekilometers per hour
Get tutor helpright-arrow

Posted 9 months ago

Question
Aliyyah is playing a game using dice. Each player takes turns rolling and adding their roll to their score.\newlineAliyyah started with −19-19 points and ended with 77 points.\newlineHow many points did Aliyyah score during the game?\newlinepoints
Get tutor helpright-arrow

Posted 9 months ago

Question
A scale on a blue print drawing of a house shows that 66 centimeters represents 33 meters.\newlineWhat number of centimeters on the blue print represents an actual distance of 2727 meters?\newlinecentimeters
Get tutor helpright-arrow

Posted 9 months ago

Question
Reem is a test driver for an automobile company. The following formula gives the total distance, d d , in feet that Reem drove a luxury car in the first t t seconds after idling at a speed of 00 miles per hour, up to the time when she passed a particular safety cone.\newlined=15.69t2 d=15.69 t^{2} \newlineCompared to the time it took Reem to pass the safety cone, how long did it take to pass a sensor that was 49 \frac{4}{9} of the distance from the start?\newlineChoose 11 answer:\newlineA) 1681 \frac{16}{81} of the time it required to pass the cone\newline(B) 49 \frac{4}{9} of the time it required to pass the cone\newline(C) 23 \frac{2}{3} of the time it required to pass the cone\newline(D) 94 \frac{9}{4} of the time it required to pass the cone
Get tutor helpright-arrow

Posted 9 months ago

Question
An ice cream cone maker wants to build a more stable cone by increasing the diameter and decreasing the height of the cone. The cone currently has diameter dd centimeters and height hh, as shown. They want the height of the new cone to be h′h'. Which of the following is closest to the smallest radius the new cone can have so that the volume is at least the volume of the old cone?
Get tutor helpright-arrow

Posted 8 months ago

Question
Victoria is shopping for thank-you notes at Sweet Greetings. She is trying to decide between the three different packages of notes shown below. \newlinePackage 11, 88 notes for $6.00\$6.00 \newlinePackage 22, 3030 notes for $12.00\$12.00 \newlinePackage 33, 1212 notes for $6.00\$6.00 \newlineIf Victoria is looking for the best deal, which package of thank-you notes should she buy?
Get tutor helpright-arrow

Posted 8 months ago

View all (14)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant