Consider a large vat containing sugar water that is to be made into soft drinks (see figure below).

Suppose:

• The vat contains 220 gallons of liquid, which never changes.
• Sugar water with a concentration of 1 tablespoons/gallon flows through pipe A into the vat at the rate of 15 gallons/minute.
• Sugar water with a concentration of 8 tablespoons/gallon flows through pipe B into the vat at the rate of 5 gallons/minute.
• The liquid in the vat is kept well-mixed.
• Sugar water leaves the vat through pipe C at the rate of 20 gallons/minute.

Let $\displaystyle {S}{\left({t}\right)}$ represent the number of tablespoons of sugar in the vat at time $\displaystyle {t}$, where $\displaystyle {t}$ is given in minutes.

(A) Write the DE model for the time rate of change of sugar in the vat:
$\displaystyle \frac{{{d}{S}}}{{{\left.{d}{t}\right.}}}=$ Preview Question 6 Part 1 of 3  


(B) Solve the differential equation to find the amount of sugar in the vat as a function of time. Your function will have an arbitrary constant $\displaystyle {K}$ in it. Assume that $\displaystyle {K}>{0}$.
$\displaystyle {S}{\left({t}\right)}=$ Preview Question 6 Part 2 of 3  


(C) Suppose that there are 40 tablespoons of sugar in the vat at $\displaystyle {t}={0}$. How many tablespoons will be present 2 minutes later?
tablespoons