Chapter 12.9, Problem 19HP

To determine

To find: The dimension of the prism that has the same surface area as the given cylinder.

Answer to Problem 19HP

The rectangular prism with length is 5.7 feet, width is 11.1 feet and height is 15 feet has the same surface area as the given cylinder. (The answer may vary).

Explanation of Solution

Given information:

The height of the cylinder is $14.6$ feet and the diameter is $10.2$ feet.

Calculation:

Write the expression to calculate the lateral area of the cylinder.

  $\text{Lateral}\text{area}=2\pi rh$

Where,

• $r$ is the radius of the cylinder.
• $h$ is the height of the cylinder.

Substitute the values in the above expression.

  $\begin{array}{c}\text{Lateral}\text{area}=2\pi \left(\frac{10.2}{2}\right)14.6\\ =468.03{\text{ft}}^2\end{array}$

Calculate the surface area of the cylinder.

  $\begin{array}{c}\text{Surface}\text{area}=\text{Lateral}\text{area}+2\pi r^2\\ =468.03{\text{ft}}^2+2\pi {\left(\frac{10.2}{2}\right)}^2{\text{ft}}^2\\ =631.52{\text{ft}}^2\\ \approx 631{\text{ft}}^2\end{array}$

Let’s consider a rectangular prism. Let the length is 5.7 feet, width is 11.1 feet and height is 15 feet.

Write the expression to calculate the surface area of the rectangular prism.

  $\text{Surface}\text{area}=2\left(lw+lh+hw\right)$

Substitute the values in the above expression.

  $\begin{array}{c}\text{Surface}\text{area}=2\left(lw+lh+hw\right)\\ =2\left(5.7\times 11.1+5.7\times 15+15\times 11.1\right)\\ =2\left(63.27+85.5+166.5\right)\\ =630.54\\ \approx 631{\text{ft}}^2\end{array}$

Conclusion:

Therefore, the rectangular prism with length is 5.7 feet, width is 11.1 feet and height is 15 feet has the same surface area as the given cylinder. (The answer may vary).