The top of a 15-foot-long ladder rests against a vertical wall with the bottom of the ladder on level ground. the ladder is sliding down the wall at a constant rate of 2 feet per second. at what rate, in radians per second, is the acute angle between the bottom of the ladder and the ground changing at the instant the bottom of the ladder is 9 feet from the base of the wall?

the rate that the acute angle changes at is -1/6 radians per second.

Step-by-step explanation:

Step 1:

From the diagram we can determine what y is by using the pythagoras theorem.

y² = h² - x²

y² = (15)² - (9)²

y = 12 ft

Step 2:

From the diagram we can see that the relationship that can help us determine the angle is the quotient of the opposite side and the adjacent side. Therefore,

tan(θ) = y/x

Now we must differentiate with respect to time on both sides.

(sec(θ))²(dθ/dt) = ( x(dy/dt) - y(dx/dt) ) / x²

sec(θ) = 1/ cos(θ)

          = 1/(9/15)

          = 5/3 rad

Step 3:

In order to determine dy/dt, we will use the pythagoras theorem.

x² + y² = h²

Now we will differentiate with respect to time.

2 x(dx/dt) + 2 y(dy/dt) = 2 h(dh/dt)

dh/dt = 0, because the length of the ladder does not change. Therefore,

2 y(dy/dt) = - 2 x(dx/dt)

dy/dt = (-2(9)(2))/ 2(12)

dy/dt = -3/2 ft/s

Step 4:

Now we have

(sec(θ))²(dθ/dt) = ( x(dy/dt) - y(dx/dt) ) / x²

(5/3)²(dθ/dt) = (9(-3/2) - 12(2)) / (9)²

dθ/dt = (9(-3/2) - 12(2)) / (9)²(5/3)²

dθ/dt = -1/6 rad/s

Therefore, the rate that the acute angle changes at is -1/6 radians per second.

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