Chapter 12.8, Problem 16PPS

To determine

To find: surface area of cylinder B

Answer to Problem 16PPS

The surface area of cylinder is $S_B=887.8m^2$

Explanation of Solution

Given:

  

Calculation:

Two cylinders, cylinder A and cylinder B of heights 5 m and 15 m respectively are similar. Also, the volume of cylinder A is 75 cubic meters, and then you have to find surface area of the cylinder B.

Since, the volume of the cylinder A is given; it can be used to find the radius of the cylinder A. Volume of cylinder

  $\begin{array}{l}A=\pi r^{2}h\\ \Rightarrow \pi r^2\left(5\right)=75\\ \Rightarrow r^2=\frac{75}{5\pi }\\ \Rightarrow r^2=4.775\\ \Rightarrow r=2.185m\end{array}$

Now, a pair of solids is similar only if their corresponding measures are proportional.

Thus, first write proportion comparing their radii and height as shown:

  $\frac{\text{Radius of cylinder B}}{\text{Radius of cylinder A}}=\frac{\text{Height of cylinder B}}{\text{Height of cylinder A}}$

Let radius of cylinder B be x.

Now, substitute the values as given,

  $\frac{x}{2.185}=\frac{15}{5}$

According to the property of the proportions, you have that if $\frac{a}{b}=\frac{c}{d}$ , then $ad=bc$ .

By this property, you have that

  $\begin{array}{l}5\left(x\right)=15\left(2.185\right)\\ \Rightarrow 5x=32.775\end{array}$

Now, divide both sides by 5 and then simplify, you will get that

  $\begin{array}{l}\Rightarrow \frac{5x}{5}=\frac{32.775}{5}\\ \Rightarrow x=6.555\text{ m}\end{array}$

Surface area of the cylinder

  $\begin{array}{l}B\left(S_B\right)=2\pi \left(6.555\right)\left(15\right)+2\pi {\left(6.55\right)}^2\\ =617.79+269.97\\ =887.76{\text{ m}}^2\end{array}$

Round of to the nearest tenths,

  $S_B=887.8m^2$

Conclusion:

Therefore, the surface area of cylinder is $S_B=887.8m^2$