Chapter 12.5, Problem 18WE

To determine

To find: The lateral area of the larger cone.

Answer to Problem 18WE

The lateral area of the larger cone is $60\pi$ .

Explanation of Solution

Given information:

The volume of similar cones are $12\pi$ and $96\pi$ .The lateral area of the smaller cone is $15\pi$ .

Calculation:

Calculate the scale factor is the cube root of ratio of volume.

  $\begin{array}{c}S=\sqrt[3]{\frac{12\pi }{96\pi }}\\ =\sqrt[3]{\frac{1}{8}}\\ =\frac{1}{2}\end{array}$

The lateral area of cone in terms of radius and slant height is $\pi rl$ .

Since lateral area of cone is proportional to the radius and slant height. Thus, the lateral area is proportional to the square of scale factor.

  $\begin{array}{c}\frac{\text{Lateral area}\text{of the}\text{smaller}\text{cone}}{\text{Lateral area}\text{of the}\text{larger}\text{cone}}={\left(\frac{1}{2}\right)}^2\\ \frac{15\pi }{\text{Lateral area}\text{of the}\text{larger}\text{cone}}=\left(\frac{1}{4}\right)\\ \text{Lateral area}\text{of the}\text{larger}\text{cone}=4\times 15\pi \\ =60\pi \end{array}$

Therefore,the lateral area of the larger coneis $60\pi$ .