Summary: ANSWER KEY
REVIEW PROBLEMS FOR MIDTERM II
MATH 2574H, SPRING 2012
UNIVERSITY OF MINNESOTA
This is a selection of problems meant to be reasonable practice for the upcoming
exam. These problems (or portions thereof) are possible types of exam questions.
Please note that the usual disclaimer continues to apply, namely, this is not guar-
anteed to be an absolutely complete review. To be perfectly ready you have to have
a grip on all the homework that has been assigned in weeks 4,5,6,7.
Salt tank problem
A tank has a huge capacity, so huge we will not worry about the possibility of
overflow. At time t = 0 the tank contains 64 gallons of water with 10 pounds of the
top secret ingredient of Calca-Cola dissolved in the water. Then solution containing
1 pound of secret ingredient per gallon of water is allowed to flow into the tank
at a rate of 5 gallons per minute. Simultaneously, well-mixed solution is allowed
to drain out of the tank at a rate of 1 gallon per minute. Find a formula for the
amount of secret ingredient in the tank at time t. Show all your work.
Let S = S(t) be the amount of secret ingredient in the tank at time t. The initial
value problem we have to solve is the following: