Chapter 12, Problem 10CR

To determine

To find: The lateral area, the total area, and the volume of the cone.

Answer to Problem 10CR

The lateral area, the total area and the volume of the cone are $60\pi {\text{cm}}^2$ , $96\pi {\text{cm}}^{\text{2}}$ ,and $96\pi {\text{cm}}^{\text{3}}$ , respectively.

Explanation of Solution

Given information:

The radius of the cone is $6\text{cm}$ and the slant height is $10\text{cm}$ .

Calculation:

Write the expression to calculate the lateral area of the cone.

  $\text{Lateral}\text{area}=\pi rl$

Where,

• $r$ is the radius of the cone.
• $h$ is the height of the cone.
• $l$ is the slant height.

Substitute the values in the above expression.

  $\begin{array}{c}\text{Lateral}\text{area}=\pi \times 6\times 10\\ =60\pi {\text{cm}}^2\end{array}$

Calculate the height of the cone.

  $\begin{array}{c}h=\sqrt{l^2-r^2}\\ =\sqrt{{10}^2-6^2}\\ =\sqrt{100-36}\\ =8\text{cm}\end{array}$

Write the expression to calculate the total area of the cone.

  $\text{Total}\text{Area}=\pi r^2+LA$

Substitute the values in the above expression.

  $\begin{array}{c}\text{Total}\text{Area}=\pi \times 6^2+60\pi \\ =36\pi +60\pi \\ =96\pi {\text{cm}}^{\text{2}}\end{array}$

Write the expression to calculate the volume of the cone.

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

Substitute the values in the above expression.

  $\begin{array}{c}V=\frac{1}{3}\times \pi \times 6^2\times 8\\ =\frac{288\pi }{3}\\ =96\pi {\text{cm}}^{\text{3}}\end{array}$

Therefore, thelateral area, the total area and the volume of the cone are $60\pi {\text{cm}}^2$ , $96\pi {\text{cm}}^{\text{2}}$ , and $96\pi {\text{cm}}^{\text{3}}$ , respectively.