A ladder is 25 feet long is leaning against the wall of the house. The base of t

Question

A ladder is 25 feet long is leaning against the wall of the house. The base of the ladder is pulling away from the wall at the rate of 2 feet per second.
a) what is the volicity of the top of the ladder when the base is given below?
1. 15 feet away from the wall ft/sec
2. 20 feet from the wall ft/sec
3. 24 feet from the wall
b) consider the triangle formed by the side of the house, ladder, and the ground. find the rate at which the area of the triangle is changing when the base of the ladder is 15 feet from the wall ft^2/sec
c) find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 15 feet from the wall rad/sec

Explanation / Answer

Assuming the ladder is infinitely thin and straight, and the ground is perfectly flat and horizontal, and the side of the house is perfectly flat and vertical, and disregarding all forces such as friction and gravity:
Let x = distance from base of ladder to house, so dx/dt = 2
Let y = distance from top of ladder to ground
By Pythagoras:
x^2 + y^2 = 25^2
x^2 + y^2 = 625
y^2 = 625 - x^2
y = sqrt(625 - x^2)
Let u = 625 - x^2, so du/dx = -2x
So y = sqrt(u) = u^0.5, so dy/du = 0.5u^(-0.5) = 1 / (2 * sqrt(625 - x^2))
By the Chain Rule:
dy/dx = dy/du * du/dx
dy/dx = (1 / (2 * sqrt(625 - x^2))) * (-2x)
dy/dx = -x / sqrt(625 - x^2)
By the Chain Rule:
dy/dt = dy/dx * dx/dt
dy/dt = (-x / sqrt(625 - x^2)) * 2
dy/dt = (-2x) / sqrt(625 - x^2)
If x = 15, then dy/dt = (-2*15) / sqrt(625 - 15^2) = -3/4 ft/sec
If x = 20, then dy/dt = (-2*20) / sqrt(625 - 20^2) = -8/3 ft/sec

If x = 24, then dy/dt = (-2*24) / sqrt(625 - 24^2) = -48/7 ft/sec

A = (1/2)xy
A = 0.5x*sqrt(625 - x^2)
Let v = 0.5x, so dv/dt = 0.5(dx/dt) = 0.5*2 = 1
So A = vy, so we use the Product Rule:
dA/dt = v(dy/dt) + y(dv/dt)
dA/dt = 0.5((-2x) / sqrt(625 - x^2)) + sqrt(625 - x^2)*1
dA/dt = ((-x) / sqrt(625 - x^2)) + sqrt(625 - x^2)

When x = 15, we have:
dA/dt = ((-15) / sqrt(625 - 15^2)) + sqrt(625 - 15^2)
dA/dt = ((-15) / sqrt(625 - 225)) + sqrt(625 - 225)
dA/dt = ((-15) / sqrt(400)) + sqrt(400)
dA/dt = (-15 / 20) + 20
dA/dt = 65/4

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