Example 3 from Algebra and Trigonometry, 5.6 Rational Functions

Solving an Applied Problem Involving a Rational Function

A large mixing tank currently contains 100 gallons of water into which 5 pounds of sugar have been mixed. A tap will open pouring 10 gallons per minute of water into the tank at the same time sugar is poured into the tank at a rate of 1 pound per minute. Find the ratio of sugar to water, in pounds per gallon in the tank after 12 minutes. Is that a greater ratio of sugar to water, in pounds per gallon than at the beginning?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Let $t$ be the number of minutes since the tap opened. Since the water increases at 10 gallons per minute, and the sugar increases at 1 pound per minute, these are constant rates of change. This tells us the amount of water in the tank is changing linearly, as is the amount of sugar in the tank. We can write an equation independently for each:

$\begin{matrix} water: W (t) & = & 100 + 10 t in gallons \\ sugar: S (t) & = & 5 + 1 t in pounds \end{matrix}$

The ratio of sugar to water, in pounds per gallon, $C$ , will be the ratio of pounds of sugar to gallons of water
$C (t) = \frac{5 + t}{100 + 10 t}$

The ratio of sugar to water, in pounds per gallon after 12 minutes is given by evaluating $C (t)$ at $t = 12.$
$\begin{matrix} C (12) & = & \frac{5 + 12}{100 + 10 (12)} \\ = & \frac{17}{220} \end{matrix}$
This means the ratio of sugar to water, in pounds per gallon is 17 pounds of sugar to 220 gallons of water.
At the beginning, the ratio of sugar to water, in pounds per gallon is
$\begin{matrix} C (0) & = & \frac{5 + 0}{100 + 10 (0)} \\ = & \frac{1}{20} \end{matrix}$
Since $\frac{17}{220} \approx 0.08 > \frac{1}{20} = 0.05,$ the ratio of sugar to water, in pounds per gallon is greater after 12 minutes than at the beginning.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

There are 1,200 freshmen and 1,500 sophomores at a prep rally at noon. After 12 p.m., 20 freshmen arrive at the rally every five minutes while 15 sophomores leave the rally. Find the ratio of freshmen to sophomores at 1 p.m.

How did it go?

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solution

Your turn

Solution

Your turn