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.Consider the triangle formed by the side of the house,the ladder,and the ground.Find the rate at which the area of the triangle is changed when the base of the ladder is 11 feet from the wall.Round your answer to two decimal places.
A)1.48
B)33.41
C)153.81
D)18.92
E)53.68

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.Consider the triangle formed by the side of the house,the ladder,and the ground.Find the rate at which the area of the triangle is changed when the base of the ladder is 11 feet from the wall.Round your answer to two decimal places. ​  ​
A)1.48 
B)33.41 
C)153.81 
D)18.92 
E)53.68

Quizplus · Accepted Answer

We want to find the rate at which the area of the triangle is changing. Let's call the base of the triangle (the side of the triangle on the ground) "x" and the height of the triangle (the distance from the ground to the top of the ladder) "h".

Using the Pythagorean Theorem, we know that:

x^2 + h^2 = 20^2

To find the area of the triangle, we need to multiply the base by the height and divide by 2:

A = (1/2) * x * h

We want to find dA/dt, the rate at which the area is changing with respect to time. We can use the product rule:

dA/dt = (1/2) * (dx/dt * h + x * dh/dt)

We know that dx/dt = 4 feet per second (since the base of the ladder is being pulled away at a constant rate), and we want to find dh/dt when x = 11 (since that is the moment in time we are interested in).

From the Pythagorean Theorem:

h^2 = 20^2 - x^2

Differentiating both sides with respect to time:

2h * dh/dt = -2x * dx/dt

Plugging in what we know:

2h * dh/dt = -2(11)(4)

h = sqrt(20^2 - 11^2) = sqrt(369)

2(sqrt(369)) * dh/dt = -2(11)(4)

dh/dt = -88/sqrt(369)

Now plugging in both dx/dt and dh/dt into our expression for dA/dt:

dA/dt = (1/2) * (4 * sqrt(369) + 11 * (-88/sqrt(369)))

Simplifying:

dA/dt = -968/sqrt(369)

Rounding to two decimal places:

dA/dt = -18.92

So the area of the triangle is decreasing at a rate of approximately 18.92 square feet per second.

A Ladder a = 20 Feet Long Is Leaning Against