A 26 foot step ladder is leaned up against a 30 foot building to reach a window near the top. The ladder is being pushed toward the building in a horizontal direction of 2 feet per second.
1. Find each related rate when the ladder is 10 feet away from the building.
a. How fast is the distance between the top of the ladder and the top of the building shrinking?
b. How fast is the angle of elevation created between the ladder and the ground changing?
c. What is the rate of change of the area of of the triangle formed between the ladder, the building, and the ground?
To solve this problem, we can use similar triangles and the concept of related rates.
First, let's define the variables:
- Let x represent the distance between the ladder and the building.
- Let h represent the height from the top of the ladder to the top of the building.
- Let θ represent the angle of elevation between the ladder and the ground.
- Let A represent the area of the triangle formed between the ladder, the building, and the ground.
Now let's solve each part of the problem:
a. To find how fast the distance between the top of the ladder and the top of the building is shrinking, we need to find dh/dt (the rate of change of h with respect to time).
We have a similar triangle formed by the ladder, the building, and the distance between them. Using the property of similar triangles, we can set up the proportion:
h / x = 30 / 26
Now, let's differentiate both sides with respect to time (t):
(dh/dt * x - h * dx/dt) / x^2 = 0
Simplifying the equation:
(dx/dt) = (dh/dt * x) / h
Substituting the given values: x = 10, h = 24, we can find dh/dt when x = 10.
b. To find how fast the angle of elevation is changing, we need to find dθ/dt (the rate of change of θ with respect to time).
We have another right triangle formed by the ladder, the distance between the ladder and the building (x), and the height (h). The tangent of θ is defined as the ratio of the opposite side (h) to the adjacent side (x):
tan(θ) = h / x
Differentiating both sides with respect to time:
sec^2(θ) * (dθ/dt) = (dh/dt * x - h * dx/dt) / x^2
Substituting the given values: x = 10, h = 24, we can find dθ/dt when x = 10.
c. To find the rate of change of the area of the triangle formed between the ladder, the building, and the ground, we need to find dA/dt (the rate of change of A with respect to time).
The area of the triangle is given by the formula:
A = (1/2) * x * h
Differentiating both sides with respect to time:
dA/dt = (1/2) * (x * dh/dt + h * dx/dt)
Substituting the given values: x = 10, h = 24, dh/dt from part (a), and dx/dt = 2, we can find dA/dt when x = 10.
By following these steps and plugging in the given values, you can find the related rates for each part of the problem when the ladder is 10 feet away from the building.