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) -(25)/(6) (B) -12 (C) -10 (D) -24

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) −256 -\frac{25}{6} \newline(B) −12-12\newline(C) −10 -\mathbf{1 0} \newline(D) −24-24

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Grade 8right chevron icon
Solve equations with variables on both sides: word problemsright 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) −256 -\frac{25}{6} \newline(B) −12-12\newline(C) −10 -\mathbf{1 0} \newline(D) −24-24
  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.
  3. Pythagorean Theorem: According to the Pythagorean 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 rates of change. We get 2xdxdt+2ydydt=02x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0, because the length of the ladder is constant, so d(392)dt=0\frac{d(39^2)}{dt} = 0.
  5. Given Rates: We know that dxdt\frac{dx}{dt}, the rate at which xx is changing, is 1010 meters per minute. We need to find dydt\frac{dy}{dt}.
  6. Finding yy: Plug in the values we know: x=36x = 36 meters, dxdt=10\frac{dx}{dt} = 10 meters/minute. We need to find yy before we can solve for dydt\frac{dy}{dt}.
  7. Calculating yy: Using the Pythagorean theorem again, we find yy by solving 362+y2=39236^2 + y^2 = 39^2. This gives us y2=392−362y^2 = 39^2 - 36^2.
  8. Solving for dydt\frac{dy}{dt}: Calculate y2y^2: y2=1521−1296y^2 = 1521 - 1296, which gives us y2=225y^2 = 225.
  9. Simplify Equation: Take the square root of y2y^2 to find yy: y=225y = \sqrt{225}, which gives us y=15y = 15 meters.
  10. Solving for dydt\frac{dy}{dt}: Now we can solve for dydt\frac{dy}{dt} using the differentiated Pythagorean theorem: 2⋅36⋅(10)+2⋅15⋅(dydt)=02\cdot 36\cdot (10) + 2\cdot 15\cdot \left(\frac{dy}{dt}\right) = 0.
  11. Final Calculation: Simplify the equation: 720+30(dydt)=0720 + 30\left(\frac{dy}{dt}\right) = 0.
  12. Final Calculation: Simplify the equation: 720+30dydt=0720 + 30\frac{dy}{dt} = 0. Solve for dydt\frac{dy}{dt}: 30dydt=−72030\frac{dy}{dt} = -720.
  13. Final Calculation: Simplify the equation: 720+30dydt=0720 + 30\frac{dy}{dt} = 0. Solve for dydt\frac{dy}{dt}: 30dydt=−72030\frac{dy}{dt} = -720. Divide both sides by 3030 to find dydt\frac{dy}{dt}: dydt=−720/30\frac{dy}{dt} = -720 / 30.
  14. Final Calculation: Simplify the equation: 720+30dydt=0720 + 30\frac{dy}{dt} = 0. Solve for dydt\frac{dy}{dt}: 30dydt=−72030\frac{dy}{dt} = -720. Divide both sides by 3030 to find dydt\frac{dy}{dt}: dydt=−720/30\frac{dy}{dt} = -720 / 30. Calculate dydt\frac{dy}{dt}: dydt=−24\frac{dy}{dt} = -24 meters/minute.

More problems from Solve equations with variables on both sides: word problems

Question
What is the amplitude of\newlineg(x)=8cos⁡(5πx+3π2)−9? g(x)=8 \cos \left(5 \pi x+\frac{3 \pi}{2}\right)-9 ? \newlineunits
Get tutor helpright-arrow

Posted 10 months ago

Question
What happens to the value of the expression 80−2r 80-2 r as r r decreases?\newlineChoose 11 answer:\newline(A) It increases.\newline(B) It decreases.\newline(C) It stays the same.
Get tutor helpright-arrow

Posted 10 months ago

Question
Solve for q q .\newline3(q+43)=2 3\left(q+\frac{4}{3}\right)=2
Get tutor helpright-arrow

Posted 9 months ago

Question
Solve for f f .\newline0.3=0.5f−0.7 0.3=0.5 f-0.7
Get tutor helpright-arrow

Posted 9 months ago

Question
Solve for a a .\newline0.25a+0.5=4.5 0.25 a+0.5=4.5
Get tutor helpright-arrow

Posted 9 months ago

Question
Solve for d d .\newline32=5d−12 \frac{3}{2}=5 d-\frac{1}{2}
Get tutor helpright-arrow

Posted 9 months ago

Question
Consider the expression 13bwh \frac{1}{3} b w h .\newlineComplete 22 descriptions of the parts of the expression.\newline11. The entire expression is the product of □\square.\newline22. In this context, 13 \frac{1}{3} is a □\square.
Get tutor helpright-arrow

Posted 9 months ago

Question
What is the sign of −8813+8813? -\frac{88}{13}+\frac{88}{13} ? \newlineChoose 11 answer:\newline(A) Positive\newline(B) Negative\newline(C) Neither positive nor negative-the sum is zero.
Get tutor helpright-arrow

Posted 9 months ago

Question
What is the sign of 3.456+(−3.456)? 3.456+(-3.456) ? \newlineChoose 11 answer:\newline(A) Positive\newline(B) Negative\newline(C) Neither positive nor negative-the sum is zero.
Get tutor helpright-arrow

Posted 9 months ago

Question
What is the sign of 123123+(−123123) 123 \frac{1}{23}+\left(-123 \frac{1}{23}\right) ?\newlineChoose 11 answer:\newline(A) Positive\newline(B) Negative\newline(C) Neither positive nor negative-the sum is zero.
Get tutor helpright-arrow

Posted 9 months ago

View all (26)

Related topics

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