#### To determine

**To find:** The rate of change of the height of the cone shape pile whose base diameter and

height are always equal.

#### Answer

The rate of change of the height of the cone shape pile is dhdt=65π≈0.38 ft/min.

#### Explanation

**Given:**

Consider the gravel is being dumped from a conveyor belt at a rate as dVdt=30 ft3/min.

The diameter of the base and height of the pile are always same.

**Formula used:**

(1) Chain rule: dydx=dydu⋅dudx

(2) Volume of the cone of base radius r and height h:V=13πr2h.

**Calculation:**

Let V be the volume of the cone shape pile and h be the height.

Since height and diameter of the cone shape pile are always same. Thus, radius of the cone shape pile is h2 .as shown in the figure-1 given below.

Since V and h changes with the time t, V and h are the function of the time t.

Obtain dhdt when h=10 ft.

Substitute r=h2 in V,

V=13π(h2)2h=112πh3

Differentiate V with respect to the time t.

ddt[V]=ddt[112πh3]dVdt=112π(3h2)dhdt [Qdydx=dydu⋅dudx]dVdt=(h24)π(dhdt)dhdt=(4πh2)dVdt

Substitute h=10 and dVdt=30 in dhdt.

dhdt=(4π×102)×30=120100π=65π≈0.38 ft/min

Therefore, the rate of change of the height of the cone shape pile is dhdt=65π≈0.38 ft/min.