Integrated Math 3 / Unit 2: Rational Expressions and Functions
2.6 Rational Models and Variation
Notes
Rational equations model rates, work, and averages.
- Work problems: if A finishes a job in $a$ hours and B in $b$ hours, together they finish in $t$ hours where $\displaystyle \frac{1}{a} + \frac{1}{b} = \frac{1}{t}$.
- Average cost $\displaystyle = \frac{\text{total cost}}{\text{number of items}}$, which has a horizontal asymptote at the per-item cost.
- Inverse variation: $\displaystyle y = \frac{k}{x}$. The product $xy$ stays constant.
Practice
Pipe A fills a tank in 4 hours. A drain empties the full tank in 12 hours. With both open, how long does it take to fill the tank?
The net rate is $\displaystyle \frac{1}{4} - \frac{1}{12} = \frac{2}{12} = \frac{1}{6}$ of the tank per hour.
A company has a $500 setup cost and $20 cost per item. Its average cost per item for $x$ items is $\displaystyle C(x) = \frac{500 + 20x}{x}$. What is $C(50)$?
$\displaystyle \frac{500 + 1000}{50} = 30$.
$y$ varies inversely with $x$, and $y = 12$ when $x = 3$. What is $y$ when $x = 9$?
$k = xy = 36$, so $\displaystyle y = \frac{36}{9} = 4$.
Ready for more? Take the Unit 2 test.