CPM Homework Help : CALC Problem 7-86

Grain pouring from a conveyor belt falls into a pile in the shape of a cone. The grain is falling at a rate of $8\frac {\text{m}^ { 3 } } { \operatorname { min } }$ , and forms a pile whose radius is always half the height. How fast is the height of the pile increasing when the radius is $6$ meters?

Step 1:

Sketch the cone, and label the sides of similar right triangles that are formed by its radius and height.

Step 2:

Write a geometric equation describing the volume of the cone in terms of height.
You will have to find a way to define $r$ in terms of $h$ .
Since the radius changes as the height changes, do NOT 'plug in' any values yet.

Step 3:

Implicitly differentiate with respect to time, $t$ .