Related rates
Example

A 10-meter ladder is leaning against the wall of a building, and the base of the ladder is sliding away from the building at a rate of 3 meters per second. How fast is the top of the ladder sliding down the wall when the base of the ladder is 6 meters from the wall?

The distance between the base of the ladder and the wall, x, and the height of the ladder on the wall, y, represent the sides of a right triangle with the ladder as the hypotenuse, h. The objective is to find dy / dt, the rate of change of y with respect to time, t, when h, x and dx / dt, the rate of change of x, are known.

Step 1:

Step 2:From the Pythagorean theorem, the equation

describes the relationship between x, y and h, for a right triangle. Differentiating both sides of this equation with respect to time, t, yields

Step 3:When solved for the wanted rate of change, dy / dt, gives us

Step 4 & 5:Using the variables from step 1 gives us:

Solving for y using the Pythagorean Theorem gives:

Plugging in 8 for the equation:

It is generally assumed that negative values represent the downward direction. In doing such, the top of the ladder is sliding down the wall at a rate of 9 / 4 meters per second.
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