Final answer:
The rate of change of the distance between the bottom of the ladder and the wall when the ladder is 7 feet from the ground is 7/8 feet per minute. Hence, option (d) is correct.
Step-by-step explanation:
The problem involves a 25-foot ladder sliding down a vertical wall at a constant rate, which is a right-angle triangle problem where the ladder serves as the hypotenuse. When the top of the ladder is 7 feet from the ground, the rate of change of the distance between the bottom of the ladder and the wall can be found using the Pythagorean theorem and related rates.
To solve, let's denote:
- y as the distance from the top of the ladder to the ground.
- x as the distance from the bottom of the ladder to the wall.
- L as the length of the ladder, which is 25 feet.
We are given that dy/dt = -3 feet per minute (the negative sign indicates the ladder is sliding down). We need to find dx/dt when y = 7 feet. From the Pythagorean theorem, we have:
L2 = x2 + y2
Differentiating both sides with respect to time t, while keeping in mind L is constant, yields:
0 = 2x(dx/dt) + 2y(dy/dt)
Solving for dx/dt:
dx/dt = -y(dy/dt) / x
Substituting the given y = 7 and dy/dt = -3:
dx/dt = -(7)(-3) / x
Using Pythagorean theorem to find x when y = 7:
x = √(L2 - y2) = √(252 - 72) = 24 feet
Therefore, dx/dt = 21 / 24 = 7/8 feet per minute.
The correct answer is d. 7/8 feet per minute.