Question

a 10-foot ladder is leaning against a wall. the bottom of the ladder is sliding away from the wall at a rate of 2ft/s. at the moment when the bottom of the ladder is 6 feet from the wall, how fast is the top of the ladder sliding down the wall?

Explanation:

Set up the geometric model

Using the Related Rates knowledge point
Let \(x(t)\) be the distance from the bottom of the ladder to the wall, and \(y(t)\) be the height of the top of the ladder on the wall.
The length of the ladder is a constant \(L = 10\text{ ft}\).
By the Pythagorean theorem:

$$x^2 + y^2 = 10^2$$

Identify given values and rates

Using the Related Rates knowledge point
At the given moment:

$$
LATEXBLOCK0
$$

Solve for the vertical height

Using the Related Rates knowledge point

$$
LATEXBLOCK1
$$

Differentiate with respect to time

Using the Implicit Differentiation knowledge point

$$
LATEXBLOCK2
$$

Calculate the rate of change of height

Using the Related Rates and Implicit Differentiation knowledge points

$$
LATEXBLOCK3
$$

The top of the ladder is sliding down the wall at a rate of \(1.5\text{ ft/s}\).

Answer:

The top of the ladder is sliding down the wall at a rate of \(1.5\text{ ft/s}\) (or \(-\frac{3}{2}\text{ ft/s}\)).