A $20$ -meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at $8$ meters per minute. $\newline$ At a certain instant, the top of the ladder is $12$ meters from the ground. $\newline$ What is the rate of change of the distance between the bottom of the ladder and the wall at that instant (in meters per minute)? $\newline$ Choose $1$ answer: $\newline$ (A) $8$ $\newline$ (B) $6$ $\newline$ (C) $\frac{32}{3}$ $\newline$ (D) $24$

Full solution

Q. A

20

-meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at

8

meters per minute.

\newline

At a certain instant, the top of the ladder is

12

meters from the ground.

\newline

What is the rate of change of the distance between the bottom of the ladder and the wall at that instant (in meters per minute)?

\newline

Choose

1

answer:

\newline

(A)

8

\newline

(B)

6

\newline

(C)

\frac{32}{3}

\newline

(D)

24

Triangle Description: We're dealing with a right triangle where the ladder is the hypotenuse, the distance from the wall is one leg, and the height above the ground is the other leg.
Pythagorean Theorem: Let's call the distance from the wall to the bottom of the ladder ＂x＂ meters. According to Pythagoras, we have $x^2 + 12^2 = 20^2$ .
Solving for x: Solving for x, we get $x^2 + 144 = 400$ , so $x^2 = 400 - 144$ .
Finding Rate of Change: Calculating that, $x^2 = 256$ , so $x = 16$ meters.
Differentiating Pythagorean Theorem: Now, we need to find the rate of change of x with respect to time, which is $\frac{dx}{dt}$ , when the height is decreasing at $8$ meters per minute.
Rate of Height Change: Differentiating both sides of the Pythagorean theorem with respect to time, we get $2x \frac{dx}{dt} + 2(12) \frac{d(12)}{dt} = 0$ .
Plugging in Values: Since the top of the ladder is sliding down at $8$ meters per minute, $\frac{d(12)}{dt} = -8$ meters per minute.
Solving for dx/dt: Plugging in the values, we get $2(16) \frac{dx}{dt} - 2(12)(8) = 0$ .
Final Rate Calculation: Solving for $\frac{dx}{dt}$ , we get $32 \frac{dx}{dt} = 192$ .
Final Rate Calculation: Solving for $\frac{dx}{dt}$ , we get $32 \frac{dx}{dt} = 192$ .Dividing both sides by $32$ , we find $\frac{dx}{dt} = 6$ meters per minute.