A 10-foot ladder is leaning straight up against a wall when a person begins pulling the base of the ladder away from the wall at the rate of 1 foot per second.

Question

What is true about the distance between the top of the ladder and the ground when the base of the ladder is 9 feet from the wall?

Answer 1

Set up an x-y axis with the wall being the y-axis and the ground being the x-axis. Let the distance from the top of the ladder to the ground be y. It is given that the ladder is 10 ft long and the base of the ladder is moving away from the wall at 1 ft/s (dx/dt). So, as the ladder is being pulled away from the way, it forms a right triangle with the ladder being the hypotenuse:

x² + y² = 100

Taking the derivative of both sides:

2xdx/dt + 2ydy/dt = 0 0r xdx/dt + ydy/dt = 0 --> dy/dt = -xdx/dt/y

When the ladder is 9 ft from the wall, 81 + y² = 100 --> y = √19 and

dy/dt = -9(1)/√19 = -(9/19)√19 or the top of the ladder is moving down at (9/19)√19 feet per second