Chapter 12, Problem 4CT

To determine

To find: The lateral area and the total area of the cone given in Exercise 3.

Answer to Problem 4CT

Thelateral area of the cone is $80\pi$ square units and the total area is $144\pi$ square units.

Explanation of Solution

Given information:

The given solid is cone with the radius 8 units,volume is $128\pi$ cubic units, and the height 6 units.

Calculation:

Write the expression to calculate the lateral area.

  $\begin{array}{c}\text{Lateral}\text{area}=\pi rl\\ =\pi r\sqrt{h^2+r^2}\end{array}$

Where,

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

Substitute $6$ for $h$ , and $8$ for $r$ in the above equation.

  $\begin{array}{c}\text{Lateral}\text{area}=\pi \times 8\sqrt{6^2+8^2}\\ =\pi \times 8\sqrt{{36}^2+64}\\ =\pi \times 8\sqrt{{100}^2}\\ =80\pi \text{square}\text{units}\end{array}$

The formula to calculate the total area of the cone is,

  $\begin{array}{c}\text{Total area}=TBA+LA\\ =\pi r^2+LA\end{array}$

Where,

• $TBA$ is the total base area of the cone.
• $LA$ is the lateral area of the cone

Substitute $80\pi$ for $LA$ , and $8$ for $r$ in the above equation.

  $\begin{array}{c}\text{Total area}=\pi 8^2+80\pi \\ =64\pi +80\pi \\ =144\pi \text{square}\text{units}\end{array}$

Therefore, the lateral area of the cone is $80\pi$ square units and the total area is $144\pi$ square units.