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.

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

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

[tex]\frac{dV}{dr}[/tex] = [tex]\frac{2}{3}[/tex][tex]\pi[/tex]rh --------------------(i)

(ii)

Given;

h = 2

r = 4

Substitute these values into equation (i) as follows;

[tex]\frac{dV}{dr}[/tex] = [tex]\frac{2}{3}[/tex][tex]\pi[/tex](4 x 2)

[tex]\frac{dV}{dr}[/tex] = [tex]\frac{2}{3}[/tex][tex]\pi[/tex](8)

[tex]\frac{dV}{dr}[/tex] = [tex]\frac{16}{3}[/tex][tex]\pi[/tex]

codiepienagoya codiepienagoya · Answer 2 · 2022-04-12T12:52:04+02:00

A right circular cone is one where the axis of cones is the line connecting the vertex to circular base's midway, the volume of right circular cone as follows:

Volume calculation:

Formula:

[tex]V = \frac{1}{3} \pi r^2h[/tex]

Where;

V = right circular cone volume

r = Cone radius.

h = Cone height.

The calculation for part 1:

[tex]\frac{dV}{dr}[/tex] is indeed the rate of change of V with reference to r when r changes but h remains constant, and it is calculated via calculating the differentiation of V with respect to r as follows:

[tex]\to \frac{dV}{dr} =\frac{d}{d}r [ \frac{1}{3} \pi r^2h] =\frac{2}{3} \pi r h[/tex]

The calculation for part 2:

When h = 2 and r = 4 then substituting the value into the part 1 equation then:

[tex]\to \frac{dV}{dr} = \frac{2}{3} \pi (4 \times 2) = \frac{2}{3} \pi (8) = \frac{16}{3} \pi[/tex]

Find out more about the volume here:

brainly.com/question/24086520