Related Rates

Applications of Derivatives: Related Rates

EXAMPLE 3: A conical container with a height of 12 inches and a radius of 6 inches is filled with water at 8 ft3/min. What is the rate of change of the height when h = 4 inches?

Start with a sketch. Do both a 3-D and a 2-D side view. Label variables.


What rates are known and unknown?

        The rate of change of volume, dV/dt = 8 ft3/min

        And dh/dt = ? when h = 4 inches

What equation relates V and h?

    V = 3.14/3 r2 h

This relates V and h, but r is a variable, too, so we   look for an equation that relates r and h. Look at the sketch.

    Using ratios:  r/h = 6/12    so r = h/2.

Substitute this into the volume equation and take the derivative with respect to t.

    V = 1.047(h/2)2h = 0.2618h3

    dV/dt = 3 (0.2618)h2 dh/dt = 0.7854h2dh/dt

Now plug in the known values and solve.