Question

(a) Write the volume V of a cone as a function of the radius r and the height h. (b) For this cone, you are given that the height is double the radius. Rewrite the function in part (a) so that it is now just a function of the volume V of a cone with respect to the height h. (substitution step)

Answer 1

Answer:

Step-by-step explanation:

The formula for calculating the volume of a cone is expressed as:

$V = \frac{\pi r^2 h}{3}$ where:

r is the radius of the cone

h is the height of the cone

Hence the volume of the cone as a function of radius r and height h is:

$V = \frac{\pi r^2 h}{3}$

b) If the height of the cone is double the radius, then:

$h = 2r$

$r = \frac{h}{2}$

Substitute $r = \frac{h}{2}$ into the formula derived in (a) as shown:

$V = \frac{\pi (\frac{h}{2} )^2 h}{3}\\ \\V = \frac{\pi (\frac{h^2}{4} ) h}{3}\\ \\V = \frac{\pi (\frac{h^3}{4} )}{3}\\ \\V = \frac{\pi h^3}{12}$

Hence function of the volume V of a cone with respect to the height h is $V = \frac{\pi h^3}{12}$