Mastery Check Solutions

Show What You Know

The surface area and volume of a cylinder are used in the efficiency ratio $\frac{S A}{V}$ .

$S A = 2 π r h + 2 π r^{2}$ $V = π r^{2} h$

Simplify the efficiency ratio for a cylinder.

$\frac{S A}{V} \frac{2 π r h + 2 π r^{2}}{π r^{2} h} = \frac{π r (2 h + 2 r)}{π r (r h)} \frac{2 h + 2 r}{r h}$

Note

Q: What is the GCF of the surface area?
A: 2πr

Once you find the GCF of the surface area, you should also see that it is also in the Volume formula. It can be simplified out of the ratio.

The restriction for the radius r is $x \neq \frac{1}{2}$ .
The restriction for the height h of the cylinder is x = 3.
Write a binomial expression for the radius and height of the cylinder.

$\begin{array}{rcl} x & = & \frac{1}{2} \\ x - \frac{1}{2} & = & 0 \\ 2 (x - \frac{1}{2}) & = & 2 (0) \\ 2 x - 1 & = & 0 \\ r & = & (2 x - 1) \end{array}$

$\begin{array}{rcl} x & = & 3 \\ x - 3 & = & 0 \\ h & = & (x - 3) \end{array}$

Note

You should work backward from the restrictions by moving all terms to one side of the equation and clearing any fractions.

The radius is (2x – 1) and the height is (x – 3).

Use parts A and B to write a simplified expression for the efficiency ratio $\frac{S A}{V}$ .

$\frac{2 h + 2 r}{r h} r = (2 x - 1) h = (x - 3)$

$\frac{2 (x - 3) + 2 (2 x - 1)}{(2 x - 1) (x - 3)} \frac{2 x - 6 + 4 x - 2}{(2 x - 1) (x - 3)} \frac{6 x - 8}{(2 x - 1) (x - 3)}$

Note

Remember to follow the order of operations when simplifying the numerator.

Say What You Know

In your own words, talk about what you have learned using the objectives for this part of the lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

- Name restrictions for a denominator.
- Simplify the product or quotient of rational expressions.