Chapter 1.3, Problem 32E

To determine

To find: the value of $x$ , the surface area and volume of the cylinder.

Answer to Problem 32E

The required result is

  $\begin{array}{l}\text{x=4}\text{.5}\\ \text{SA=183}\text{.2}\\ \text{V=183}\text{.2}\end{array}$

Explanation of Solution

Given:

  

Calculation:

Surface area and volume formulas for a cylinder:

  $\begin{array}{l}SA=2\pi r^2+2\pi rh\\ V=\pi r^{2}h\end{array}$

Find the height:

  $2\pi r^2+2\pi rh=\pi r^{2}h$ Set the equations equal to the each other.

  $\text{2}\pi {\left(3.6\right)}^2+2\pi \left(3.6\right)\left(x\right)=\pi {\left(3.6\right)}^2\left(x\right)$ Substitute the radius and height into the formula.

  $\text{2}\pi \left(12.96\right)+2\pi \left(3.6\right)\left(x\right)=\pi \left(12.96\right)\left(x\right)$ Evaluate exponents.

  $25.92\pi +7.2\pi x\text{=12}\text{.96}\pi \text{x}$ Multiply.

  $\text{25}\text{.92}\pi \text{=5}\text{.76}\pi \text{x}$ Subtract $\text{7}\text{.2}\pi \text{x}$ from both sides.

  $\text{4}\text{.5= x}$ Divide both sides by $\text{5}\text{.76}\pi .$

Find the surface area:

  $\text{SA=2}\pi {\left(3.6\right)}^2\text{+2}\pi \left(3.6\right)\left(4.5\right)$ Substitute the radius and height into the formula.

  $\text{SA=2}\pi \left(12.96\right)+2\pi \left(3.6\right)\left(4.5\right)$ Evaluate exponents.

  $\text{SA=25}\text{.92}\pi \text{+32}\text{.4}\pi$ Multiply

  $\text{SA=58}\text{.32}\pi$ Simplify.

  $\text{SA=183}\text{.2 }$ Multiply.

Find the volume:

  $\text{V=}\pi {\left(3.6\right)}^2\left(4.5\right)$ Substitute the radius and height into the formula.

  $\text{V=}\pi \left(12.96\right)\left(4.5\right)$ Evaluate exponents.

  $\text{V=58}\text{.32}\pi$ Multiply.

  $\text{V=183}\text{.2}$ Multiply.

Thus,

  $\begin{array}{l}\text{x=4}\text{.5}\\ \text{SA=183}\text{.2}\\ \text{V=183}\text{.2}\end{array}$

Conclusion:

Hence the result is:

  $\begin{array}{l}\text{x=4}\text{.5}\\ \text{SA=183}\text{.2}\\ \text{V=183}\text{.2}\end{array}$