The volume of a cone is 141.3 cubic inches. The height of the cone is 15 inches What is the radius of the cone, rounded to the nearest inch?

Dec 24, 2017

See a solution process below:

Explanation:

The formula for the Volume of a cone is:

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

Where:

$V$ is the Volume of the cone: $141.3 {\text{in}}^{3}$ for this problem.

$r$ is the radius of the cone: what we are solving for in this problem.

$h$ is the height of the cone: $15 \text{in}$ for this problem.

Substituting and solving for $r$ gives:

141.3"in"^3 = pi xx r^2 xx (15"in")/3

$141.3 \text{in"^3 = pi xx r^2 xx 5"in}$

$\left(141.3 \text{in"^3)/color(red)(5"in") = (pi xx r^2 xx 5"in")/color(red)(5"in}\right)$

(141.3"in"^(color(red)(cancel(color(black)(3)))2))/color(red)(5color(black)(cancel(color(red)("in")))) = (pi xx r^2 xx color(red)(cancel(color(black)(5"in"))))/cancel(color(red)(5"in"))

$\frac{141.3 {\text{in}}^{2}}{\textcolor{red}{5}} = \pi {r}^{2}$

$28.26 {\text{in}}^{2} = \pi {r}^{2}$

$\frac{28.26 {\text{in}}^{2}}{\textcolor{red}{\pi}} = \frac{\pi {r}^{2}}{\textcolor{red}{\pi}}$

$\frac{28.26 {\text{in}}^{2}}{\textcolor{red}{\pi}} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{\pi}}} {r}^{2}}{\cancel{\textcolor{red}{\pi}}}$

$\frac{28.26 {\text{in}}^{2}}{\textcolor{red}{\pi}} = {r}^{2}$

We can use 3.1416 to estimate $\pi$ giving:

$\frac{28.26 {\text{in}}^{2}}{\textcolor{red}{3.1416}} = {r}^{2}$

$9 {\text{in}}^{2} = {r}^{2}$ rounded to the nearest inch.

Now, take the square root of each side of the equation to find the radius of the cone while keeping the equation balanced:

$\sqrt{9 {\text{in}}^{2}} = \sqrt{{r}^{2}}$

$3 \text{in} = r$

$r = 3 \text{in}$

The radius of the cone rounded to the nearest inch is 3 inches.