Related Rates

Example 1: If the radius of a circle is increasing at the rate of 7cm/sec, how fast is its area increasing when the radius is 20 cm? r r r

Solution to Example 1: Let A be the area and r the radius of circle at time t. Know Find

By the chain rule,

Example 2: One end of a 13-foot ladder is on the ground and the other end rests on a vertical wall. If the bottom end slides away from the wall at the rate of 3 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 5 feet from the wall?

Differentiating implicitly w.r.t. x, we get

Example 3: An object is moving clockwise around the circle As it passes through the point the y-coordinate is decreasing at the rate of 3 units per second. At what rate is its x-coordinate changing at that point?

Solution to Example 3: Know Find

Example 4: A conical paper cup 8 cm across the top and 12 cm deep is full of water. Water begins to leak out of the bottom at the rate of How fast is the level of water dropping when the water is 3 cm deep?

Let V, r, and h denote the volume, radius, Solution to Example 4: Let V, r, and h denote the volume, radius, and hight, respectively, of the water remaining at time t Know Find

Example 5: The area of an equilateral triangle is increasing at a rate of Find the rate the length of each side is increasing when the area is

Solution to Example 5: Let A, x, and h denote the area, length of each side, and the height, respectively, of the triangle at time t min. Know Find