Suppose a 170-ft ladder is leaning against a wall. If the bottom is pulled away from the wall at a rate of 1 ft/sec, at what rate is the top of the ladder sliding down the wall when the bottom is 26 ft from the wall?
A)The top of the ladder is sliding down the wall at the rate of Ft/sec.
B)The top of the ladder is sliding down the wall at the rate of Ft/sec.
C)The top of the ladder is sliding down the wall at the rate of Ft/sec.
D)The top of the ladder is sliding down the wall at the rate of Ft/sec.
E)The top of the ladder is sliding down the wall at the rate of Ft/sec.

Question

Suppose a 170-ft ladder is leaning against a wall. If the bottom is pulled away from the wall at a rate of 1 ft/sec, at what rate is the top of the ladder sliding down the wall when the bottom is 26 ft from the wall? ​
A)The top of the ladder is sliding down the wall at the rate of Ft/sec. 
B)The top of the ladder is sliding down the wall at the rate of Ft/sec. 
C)The top of the ladder is sliding down the wall at the rate of Ft/sec. 
D)The top of the ladder is sliding down the wall at the rate of Ft/sec. 
E)The top of the ladder is sliding down the wall at the rate of Ft/sec.

Quizplus · Accepted Answer

To solve this problem, we need to use related rates. Let's label the sides of the right triangle formed by the wall, the ground, and the ladder.

- Let x be the distance from the ladder's base on the ground to the wall.
- Let y be the height of the ladder on the wall.
- Let z be the length of the ladder (which we know is 170 ft).

From the Pythagorean theorem, we know that:

x^2 + y^2 = z^2

Now, we'll implicitly differentiate this equation with respect to time (t) using the chain rule:

2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)

Note that we don't know dx/dt or dy/dt, but we do know that dx/dt = -1 (since we're told that the bottom of the ladder is being pulled away from the wall at a rate of 1 ft/sec). We're also given that z = 170 ft. Finally, we need to find dy/dt when x = 26 ft.

We can plug in all our known values and solve for dy/dt:

2(26)(-1) + 2y(dy/dt) = 2(170)(0)

-52 + 2y(dy/dt) = 0

2y(dy/dt) = 52

dy/dt = 52/(2y)

At this point, we need to find y when x = 26. We can use the Pythagorean theorem to do this:

26^2 + y^2 = 170^2

y^2 = 170^2 - 26^2 = 27,446

y = sqrt(27,446) = 165.6 (rounded to one decimal place)

Now we can plug in y and solve for the rate at which y is changing (i.e. dy/dt):

dy/dt = 52/(2y) = 52/(2*165.6)

dy/dt = 0.157 ft/sec

So the top of the ladder is sliding down the wall at a rate of approximately 0.157 ft/sec when the bottom is 26 ft from the wall.

Therefore, the best choice is C.

Suppose a 170-Ft Ladder Is Leaning Against a Wall