Question

A large mixing tank currently contains $200$ gallons of water into which $10$ pounds of sugar have been mixed. A tap will open pouring $20$ gallons per minute of water into the tank at the same time sugar is poured into the tank at a rate of $2$ pound per minute. Find the concentration (pounds per gallon) of sugar in the tank after $14$ minutes. Then

• A. the concentration is greater than at the beginning?
• B. the concentration lesser than at the beginning?
• C. the concentration equal to the concentration at the beginning?
• D. None of the above

Answer 1

The correct option is A the concentration is greater than at the beginning?

Let $t$ be the number of minutes since the tap opened.

Since the water increases at $20$ gallons per minute, and the sugar increases at $2$ 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:

$\text{Water}:W\left(t\right)=200+20t$ in gallons

$\text{Sugar}:S\left(t\right)=10+2t$ in pounds​​

The concentration $C$ will be the ratio of pounds of sugar to gallons of water.

$C\left(t\right)=\frac{10+2t}{200+20t}$​​

The concentration after $14$ minutes is given by evaluating $C\left(t\right)$ at $t=14$

$\therefore C\left(14\right)=\frac{10+28}{200+280}=\frac{19}{240}$

This means the concentration is $19$ pounds of sugar to $240$ gallons of water.

At the beginning, the concentration is $C\left(0\right)=\frac{10+0}{200+0}=\frac{1}{20}$.

Since $\frac{19}{240}\approx 0.08>\frac{1}{20}=0.05$, the concentration is greater after $14$ minutes than at the beginning.