Chapter 12.8, Problem 25HP

To determine

To Describe: the surface area of a cone.

Answer to Problem 25HP

The surface area quadruples if radius and slant height are doubled.

Explanation of Solution

Given:

Radius and slant height are doubled.

Calculation:

Since the two cones are similar and a pair of solids is similar only if their corresponding measures are proportional.

Since, the dimensions of the new cone are the double of the original cone.

Hence, the scale factor is $\frac{1}{2}$ .

If two solids are similar with a scale factor of $\frac{a}{b}$ , then their surface areas have the ratio ${\left(\frac{a}{b}\right)}^2$ .

That is,

  $\begin{array}{l}\frac{Surfaceareaoftheoriginalcone}{Surfaceareaofthenewcone}={\left(\frac{a}{b}\right)}^2\\ ={\left(\frac{1}{2}\right)}^2\\ =\frac{1}{4}\end{array}$

Hence, the surface area of new cone $=4$ (Surface area of original cone)

Thus, the surface area quadruples.

Conclusion:

Thus, the surface area quadruples.