Unit 2 Lesson 16 Practice Problem 2

(Note: Type any negative numbers without a space between the sign and the number, e.g., -17. Do NOT include commas for spacing between sets of three digits, e.g., 250000. Include a 0 before the decimal point when necessary, e.g., 0.2)

The surface area [tex]$ S(r) $[/tex] in square units of a cylinder with a volume of 18 cubic units is a function of its radius [tex]$ r $[/tex] in units, where:

[tex]\[ S(r) = 2\pi r^2 + \frac{35}{r} \][/tex]

What is the surface area of a cylinder with a volume of 18 cubic units and a radius of 3 units? Round to two decimal places.

[tex]$ S(3) = $[/tex]

To find the surface area of a cylinder with a given volume of 18 cubic units and a radius of 3 units, we will use the provided surface area formula for the cylinder:

[tex]\[ S(r) = 2 \pi r^2 + \frac{35}{r} \][/tex]

Here, [tex]$ r = 3 $[/tex] units. Let's calculate the surface area step-by-step:

1. Calculate [tex]$ 2 \pi r^2 $[/tex]:
- First, find [tex]$ r^2 $[/tex]. Since [tex]$ r = 3 $[/tex]:
[tex]\[
r^2 = 3^2 = 9
\][/tex]
- Next, multiply by [tex]$ 2 \pi $[/tex]:
[tex]\[
2 \pi \times 9 = 18\pi
\][/tex]

2. Calculate [tex]$\frac{35}{r}$[/tex]:
- Substitute [tex]$ r = 3 $[/tex]:
[tex]\[
\frac{35}{3} \approx 11.6667
\][/tex]

3. Combine the terms to find [tex]$ S(3) $[/tex]:
- Add the results from the above steps:
[tex]\[
S(3) = 18\pi + \frac{35}{3}
\][/tex]
- Use the approximation [tex]$ \pi \approx 3.14159 $[/tex] to calculate:
[tex]\[
18 \times 3.14159 \approx 56.5487
\][/tex]
- Add the result of [tex]$\frac{35}{3}$[/tex]:
[tex]\[
56.5487 + 11.6667 \approx 68.2154
\][/tex]

4. Round the result to two decimal places:
- The surface area [tex]$ S(3) $[/tex] is approximately:
[tex]\[
S(3) \approx 68.22
\][/tex]

So, the surface area of the cylinder, rounded to two decimal places, is [tex]$ 68.22 $[/tex] square units.