Question

For a right circular cone, if the radius is reduced to 1/5 its original size and the slant height is reduced to 1/6 its original size, find the surface area if the original radius is 8 centimeters and the original slant height is 13 centimeters.

Answer 1

Check the picture below.

$\bf \begin{cases} \underline{r=8}\\ \frac{1}{5}r\implies \frac{1}{5}\cdot 8\\ r=\frac{8}{5}\\ --------\\ z=slant\ height\\ \underline{z=13}\\ \frac{1}{6}z\implies \frac{1}{6}\cdot 13\\ z=\frac{13}{6} \end{cases}\qquad \begin{array}{llll} S=\pi rz+\pi r^2 \\\\\\ S=\pi \cdot \cfrac{8}{5}\cdot \cfrac{13}{6}+\pi \cdot \left( \cfrac{8}{5} \right)^2 \end{array} \\\\\\ S=\cfrac{104\pi }{30}+\cfrac{8^2}{5^2}\implies S=\cfrac{104\pi }{30}+\cfrac{64}{25}\implies S=\cfrac{520\pi +384}{150}$