Question

The volume of a right circular cone of radius r and height h is V = 1 3 πr 2h (a) (i) Find a formula for the instantaneous rate of change of V with respect to r if r changes and h remain constant. (ii) Suppose that h = 2 is fixed but r varies. Find the rate of change of V w. R. To r at the point where r = 4.

Answer 1

Answer:

(i) $\frac{2}{3}$$\pi$rh

(ii) $\frac{16}{3}$$\pi$

Explanation:

Given:

V = $\frac{1}{3}$$\pi$r²h

Where;

V = volume of a right circular cone.

r = radius of the cone

h = height of the cone.

(i) The rate of change of V with respect to r if r changes and h remains constant is $\frac{dV}{dr}$, and is given by finding the differentiation of V with respect to r as follow:

$\frac{dV}{dr}$ = $\frac{d}{dr}$[$\frac{1}{3}$$\pi$r²h]

$\frac{dV}{dr}$ = $\frac{2}{3}$$\pi$rh     --------------------(i)

(ii)

Given;

h = 2

r = 4

Substitute these values into equation (i) as follows;

$\frac{dV}{dr}$ = $\frac{2}{3}$$\pi$(4 x 2)

$\frac{dV}{dr}$ = $\frac{2}{3}$$\pi$(8)

$\frac{dV}{dr}$ = $\frac{16}{3}$$\pi$

$\frac{dV}{dr}$ = $\frac{16}{3}$$\pi$