Chapter 12.5, Problem 11WE

To determine

To find:The total area of the larger cone.

Answer to Problem 11WE

The total area of the larger coneis $64\pi {\text{cm}}^{\text{2}}$ .

Explanation of Solution

Given information:

The radii of two similar cones are $4\text{cm}$ and $6\text{cm}$ . The area of smaller cone is $36\pi {\text{cm}}^2$ .

Calculation:

The ratio of the corresponding sides of any two similar geometric figures is termed as Scale factor.

Let the total area of the smaller cone be $t{a}_1$ and the total area of the larger cone be $t{a}_2$ , and scale factor here for both the cones be $r_1$ and $r_2$ .

The theorem of similar solids states that if the scale factor of two similar solids is $a:b$ , then the ratio of base areas, lateral areas and total areas is $a^2:b^2$ .

Expression to determine the total area of two similar cones from the above theorem is:

  $\begin{array}{c}\text{total}\text{area}={\left(\text{scale}\text{factor}\right)}^2\\ \frac{t{a}_1}{t{a}_2}={\left(\frac{r_1}{r_2}\right)}^2\end{array}$

Substitute $4$ for $r_1$ , $6$ for $r_2$ and $36\pi {\text{cm}}^2$ for $t.a_1$ in the above expression.

  $\begin{array}{c}\frac{36\pi }{t{a}_2}={\left(\frac{3}{4}\right)}^2\\ \frac{36\pi }{t{a}_2}=\frac{9}{16}\\ t{a}_2=\frac{36\pi \times 16}{9}\\ =64\pi \end{array}$

Therefore, the total area of the larger cone is $64\pi {\text{cm}}^{\text{2}}$ .