Question

a 15 foot ladder is resting against the wall. the bottom is initially 10 feet away from the wall and is being pushed towards the wall at a rate of 14 ft/sec. how fast is the top of the ladder moving up the wall 12 seconds after we start pushing?

Answer 1

The top of the ladder is moving down the wall at a rate of 9.33 ft/sec after 12 seconds of pushing.

Step-by-step explanation:

To determine how fast the top of the ladder is moving up the wall, we can use related rates. We know that the distance between the bottom of the ladder and the wall is decreasing at a rate of 14 ft/sec.

Let's call this distance x. The height of the ladder can be considered as y, and based on the given information, it remains constant at 15 feet.

Using the Pythagorean theorem, we have x^2 + y^2 = 15^2, where x represents the distance between the bottom of the ladder and the wall.

By differentiating both sides of the equation with respect to time, we get 2x(dx/dt) + 2y(dy/dt) = 0.

Since we want to find dy/dt (the rate at which the top of the ladder is moving up the wall), we can rearrange the equation to solve for dy/dt, which gives us dy/dt = -x(dx/dt) / y.

Now, plugging in the given values, we have -10(14) / 15 = -140/15

= -28/3 feet per second.

Therefore, the top of the ladder is moving down the wall at a rate of 28/3 ft/sec after 12 seconds of pushing.