A ladder a = 20 feet long is leaning against the wall of a house (see figure).The base of the ladder is pulled away from the wall at a rate of b = 4 feet per second.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 19 feet from the wall.Round your answer to three decimal places.

Question

A ladder a = 20 feet long is leaning against the wall of a house (see figure).The base of the ladder is pulled away from the wall at a rate of b = 4 feet per second.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 19 feet from the wall.Round your answer to three decimal places. ​  ​​

Quizplus · Accepted Answer

Let θ be the angle between the ladder and the wall. Then we have:

cos(θ) = adjacent/hypotenuse = x/20

Differentiating both sides with respect to time t, we get:

-sin(θ) dθ/dt = (1/20)dx/dt

We are given that dx/dt = 4 ft/s and x = 19 ft (since the ladder remains 20 ft long). Thus:

sin(θ) dθ/dt = -(1/20)(4)

We need to find dθ/dt when x = 19 ft. To do this, we need to find sin(θ) when x = 19 ft. Using the equation above:

cos(θ) = x/20 = 19/20

sin(θ) = √(1 - cos²(θ)) = √(1 - 361/400) = √(39/400) = √39/20

Substituting sin(θ) and dx/dt into the equation above, we get:

(√39/20) dθ/dt = -(1/5)

Therefore:

dθ/dt = -(1/5)(20/√39) = -4√39/39

Rounding to three decimal places:

dθ/dt = -0.641 ft/s

Thus, the answer is B).

A Ladder a = 20 Feet Long Is Leaning Against