Final answer:
To find how fast the top of the ladder is moving down the wall, we differentiate the Pythagorean theorem with respect to time. Using the values given, we find that dy/dt = -7/12 feet/second, which means the ladder's top is moving down the wall at a rate of 7/12 feet per second.
Step-by-step explanation:
To determine how fast the top of the ladder is moving down the wall when the base is sliding away at a rate of 2 feet/second, we can use the Pythagorean theorem in a right-angled triangle formed by the wall, the floor and the ladder. Let's denote the distance of the base of the ladder from the wall as x, the height the ladder reaches up the wall as y, and the length of the ladder as L, which is a constant 25 feet.
Applying the Pythagorean theorem, we have:
- x² + y² = L²
Given that L is constant, we can differentiate both sides of the equation with respect to time t:
- 2x(dx/dt) + 2y(dy/dt) = 0
We are given that dx/dt is 2 feet/second when x =7 feet. We need to find y at this instance using the Pythagorean theorem:
- 7² + y² = 25²
- y² = 625 - 49
- y = √576
- y = 24 feet
We substitute x = 7, dx/dt = 2, and y = 24 into the differentiated Pythagorean theorem to solve for dy/dt:
- 2*7*(2) + 2*24(dy/dt) = 0
- 28 + 48(dy/dt) = 0
- 48(dy/dt) = -28
- dy/dt = -28/48
- dy/dt = -7/12 feet/second
Therefore, the top of the ladder is moving down at the rate of -7/12 feet/second. The negative sign indicates that the top of the ladder is moving downward along the wall.