Final answer:
The question asks for the rate at which a ladder is sliding down a wall when the top is 8 feet from the ground, given it is sliding away from the bottom at 4 ft/sec. By applying related rates and the Pythagorean theorem, we can derive and solve an equation to find this rate of descent.
Step-by-step explanation:
The question deals with a ladder that is sliding down a wall. The scenario here is a classic physics problem that involves related rates, which are rates at which related variables change with respect to time. We are given that the ladder is 20 feet long and slides away from the wall at a speed of 4 ft/sec. We need to find out how fast the ladder is sliding down when the top is 8 feet off the ground.
To solve this, we can apply the Pythagorean theorem to a right-angled triangle formed by the ladder, the wall, and the ground. Let's call the distance from the base of the ladder to the wall x, the height the ladder reaches up the wall y, and the length of the ladder L. We are told that L is 20 feet, so we have L² = x² + y². Differentiating both sides with respect to time t, we get 2x(dx/dt) + 2y(dy/dt) = 0. We can solve for dy/dt which will tell us the speed at which the top of the ladder is sliding down.
Plugging in the given rates and distances, when y = 8 feet, we can solve for x using Pythagorean theorem to be x = √(L² - y²) = √(20² - 8²) feet. Differentiating implicitly gives us dy/dt = -(x/y)(dx/dt). Substituting x, y, and dx/dt (4 ft/sec) into the equation, we can find the value for dy/dt, which is the rate at which the ladder slides down the wall.