Question

The formula for the volume of a cone is V = 1/3pir2h where V=volume, r=radius of the base of the cone, and h=height of the cone.

Part A. Rewrite the formula for the volume of a cone in terms of its radius.

Part B: A cone has a volume of 25 cubic meters and a height of 5 meters. Find the exact value for r, the radius of the cone.

Answer 1

Part A:

Let's rewrite the equation $V = \frac{1}{3} \pi r^2 h$ in terms of $r$:

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

• Set up equation

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

• Divide both sides by π and multiply both sides by 3

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

• Divide both sides by $h$

$r = \pm \sqrt{\dfrac{3V}{h \pi}}$

• Square root both sides of the equation

$\boxed{r = \sqrt{\dfrac{3V}{h \pi}}}$

• $r = -\sqrt{\dfrac{3V}{h \pi}}$ is an extraneous solution, as we cannot have a negative radius for a cone

Part B:

Let's use the equation we just found to find $r$:

$r = \sqrt{\dfrac{3V}{h \pi}}$

• Set up equation

$r = \sqrt{\dfrac{3(25)}{5 \pi}} = \sqrt{\dfrac{75}{5 \pi}} = \sqrt{\dfrac{15}{\pi}}$

• Input values and simplify

Our answer is: $\boxed{r = \sqrt{\dfrac{15}{\pi}}}$