Question

Which of the following functions gives the radius, r(v), of a conical artifact that is 20 inches tall as a function of its volume, v, in cubic inches?
A.

B.

C.

D.

Answer 1

Radius as function of volume is:

$r(v) = \sqrt{\frac{3v}{20 \pi }}\ inches$

Solution:

The volume of cone is given as:

$v = \frac{1}{3}\pi r^{2} h$

Where,

r is the radius

h is the height

From given,

height = 20 inches

From formula,

$v = \frac{1}{3}\pi r^{2} h$

Rearrange , so that r is alone in left side of equation

$3v = \pi r^2 h\\\\\pi r^2 h = 3v\\\\r^2 = \frac{3v}{\pi h}\\\\Take\ square\ root\ on\ both\ sides\\\\r = \sqrt{\frac{3v}{\pi h}}$

Substitute h = 20

$r = \sqrt{\frac{3v}{20 \pi }}$

Thus, radius as function of volume is:

$r(v) = \sqrt{\frac{3v}{20 \pi }}\ inches$