Chapter 12.3, Problem 3CE

(a)

To determine

To find: The total areas of cylinders $\text{I}$ , $\text{II}$ and $\text{III}$ .

Answer to Problem 3CE

The total areas of cylinders $\text{I}$ , $\text{II}$ and $\text{III}$ are $42\pi ,66\pi ,120\pi$ respectively.

Explanation of Solution

Given information:

The given figure is shown below

  

Figure (1)

Calculation:

Total area of cylinder is given

  $2\pi rh+2\pi r^2=2\pi r\left(r+h\right)$

Total areas of cylinders are as follows,

The values for cylinder $\text{I}$ .

  $\begin{array}{l}r=3\\ h=4\end{array}$

Calculate the total area for cylinder $\text{I}$ .

  $\begin{array}{c}2\pi r\left(r+h\right)=2\pi \times 3\left(3+4\right)\\ =42\pi \end{array}$

The values for cylinder $\text{II}$ .

  $\begin{array}{l}r=3\\ h=8\end{array}$

Calculate the total area for cylinder $\text{II}$ .

  $\begin{array}{c}2\pi r\left(r+h\right)=2\pi \times 3\left(3+8\right)\\ =66\pi \end{array}$

The values for cylinder $\text{III}$ .

  $\begin{array}{l}r=6\\ h=4\end{array}$

Calculate the total area for cylinder $\text{III}$ .

  $\begin{array}{c}2\pi r\left(r+h\right)=2\pi \times 6\left(6+4\right)\\ =120\pi \end{array}$

Therefore, thetotal areas of cylinders $\text{I}$ , $\text{II}$ and $\text{III}$ are $42\pi ,66\pi ,120\pi$ respectively.

b)

To determine

To find: Whether the ratios of the total areas are same as those of lateral areas in exercise 2.

Answer to Problem 3CE

The ratio of the total areas and lateral areas are not equal.

Explanation of Solution

Given information:

The ratio of the total area of cylinder $\text{I}$ , $\text{II}$ and $\text{III}$ respectively are.

  $42\pi :66\pi :120\pi$

Calculation:

The ratio of the total area of cylinder $\text{I}$ , $\text{II}$ and $\text{III}$ respectively are.

  $42\pi :66\pi :120\pi$

The ratio of the lateral area of cylinder $\text{I}$ , $\text{II}$ and $\text{III}$ respectively are.

  $24\pi :48\pi :48\pi$

From above calculation it can be seen that the ratio of the total areas and lateral areas are not equal.

Therefore, the ratio of the total areas and lateral areas are not equal.