Question

The lateral surface area Sof a cone can be found by using the formula $S=\pi r\sqrt{r^2+h^2}$, where r is the radius of the base and h is the height of the cone. Find the height of the cone.

Short answer

The height of the given cone is$12.48\text{in.}$

Step-by-step solution

Step 1. Write the formula for the lateral surface area of the cone.

The lateral surface area (S) of a cone is given by:

$S=\pi r\sqrt{r^2+h^2}$, where r is the radius of the cone and h is the height of the cone.

Step 2. Observe the given diagram.

The given diagram is:

From the given diagram, it can be noticed that the lateral surface area (S) of a cone is$121{\text{in}}^2$ and the radius of the cone (r) is 3 in.

Step 3. Find the height of the cone.

Substitute 121 for S and 3 for r in the equation$S=\pi r\sqrt{r^2+h^2}$ and solve for h.

$$\begin{gathered} S=\pi r\sqrt{r^2+h^2} \\ 121=\pi \left(3\right)\sqrt{3^2+h^2} \\ \frac{121}{3\pi }=\sqrt{9+h^2} \\ 12.84=\sqrt{9+h^2} \\ {\left(12.84\right)}^2={\left(\sqrt{9+h^2}\right)}^2 \\ 164.8656=9+h^2 \\ 164.8656-9=h^2 \\ 155.8656=h^2 \\ \pm \sqrt{155.8656}=h \\ \pm 12.48=h \end{gathered}$$

As, the height of the cone cannot be negative, therefore$h=-12.48$ is not possible.

The only possible value h of is $12.48$.

Therefore, the height of the cone is$12.48$ in.