Chapter 12.8, Problem 3CYU

To determine

Find the ratio of the surface area of the small cylinder to the surface area of the large cylinder.

Answer to Problem 3CYU

4:25

Explanation of Solution

Given:

Two similar cylinders have radii of 15 inches and 6 inches .

Formula Used:

Two solids are similar if they have the same shape and the ratios of their corresponding linear measures are equal.

The Surface area of a cylinder is :

  $S=2\pi rh+2\pi r^2$ ; where r is the radius and h is the height of the cylinder.

Calculation:

Given:

  $\begin{array}{l}{r}_1=6inches\\ r_2=15inches\end{array}$

Since the two cylinders are similar , the ratio of their corresponding linear measures are equal:

  $\Rightarrow \frac{r_1}{r_2}=\frac{h_1}{h_2}=\frac{6}{15}=\frac{2}{5}=k$ ; where $h_1$ is the height of the small cylinder and $h_2$ is the height of the larger cylinder.

So,

The ratio of the surface area of the smaller cylinder to the larger cylinder is :

  $\begin{array}{l}\frac{S_1}{S_2}\\ =\frac{2\pi r_1{h}_1+2\pi r_1{}^2}{2\pi r_2{h}_2+2\pi r_2{}^2}\\ =\frac{2\pi r_1\left(h_1+r_1\right)}{2\pi r_2\left(h_2+r_2\right)}\\ =\frac{r_1\left(h_1+r_1\right)}{r_2\left(h_2+r_2\right)}\\ =\frac{r_1}{r_2}\cdot \frac{\left(h_1+r_1\right)}{\left(h_2+r_2\right)}\\ =k\cdot \frac{\left(k{h}_2+k{r}_2\right)}{\left(h_2+r_2\right)}\\ =k\cdot \frac{k\left(h_2+r_2\right)}{\left(h_2+r_2\right)}\\ =k^2\\ ={\left(\frac{2}{5}\right)}^2\\ =\frac{4}{25}\end{array}$

So, the ratio of the surface area of the small cylinder to the surface area of the large cylinder is 4:25.