Math Help

Examples of Related Rates

Example 1: Assume that the infected area of an injury is circular. If the radius of the

infected area is 3 mm and growing at a rate of 1 mm/hr, at what rate is the infected area

increasing?

r = r( t )

A = area = π r², dA/dt = 2 π r dr/dt

Evaluate dA/dt when the radius is 3mm and the growth rate is 1 mm/hr.

So dA/dt | _{r = 3} = 2 π (3)(1) = 6 π mm² / hr (result)

Example 2: A dock is 6 feet above the water (h = 6 ft). Suppose you stand on the

edge of the dock and pull a rope attached to the boat at the constant rate of 2 ft/sec.

Assume that the boat remains at water level. Find the speed of the boat when it is

20 feet from the dock. Also find its speed when it is 10 feet from the dock. Does the

speed of the boat remain constant?

Apply the Pythagorean theorem.

x² + h² = L² where h = 6 ft Now apply implicit differentiation.

2x dx/dt = 2L dL/dt, so dx/dt = (L/x) dL/dt

For x = 20 ft, L = √ ( 20² + 6² ) so dx/dt = (√436 /20 ) (2) = 2.08 ft/sec

For x = 10 ft, L = √ ( 10² + 6² ) so dx/dt = (√136 /20 ) (2) = 2.33 ft/sec

Result: Speed of boat increases as it gets closer to dock.