Midterm 3, Math 30, Fall 2008, 12/08/08
Instructions: Write your name and section number. Draw grading table on the cover.
Read each problem carefully and follow all of its instructions. For each of the problems
below, write a clear and concise solution in your blue book. Solutions must be simplified
as much as possible, no full credit for partially completed problems. Blue books with
torn or missing pages will not be accepted !
1. (10 pts) Solve the differential equation 3=+ x
y
dx
dy and 2
)1( =y
2. (10 pts) The population of aphids on a rose plant increases at the rate
proportional to the number present. After 1 day the population increased
from 100 to 101. How long will it take for population to get to 1000?
3. Write down but don’t solve differential equations for following problems.
Let the proportionality constant be k > 0.
a. (5 pts) A glucose solution is administered intravenously into the
bloodstream at a constant rate r. The glucose is removed from the
bloodstream at a rate proportional to the concentration at that time.
Write down the differential equation for concentration over time
dC/dt where C is the concentration at time t.
b. (5 pts) A hot object cools off at the rate proportional to the
temperature difference between the object and surrounding air. Let
T(t) be the temperature of the object and TA be the ambient
temperature of surrounding air (T > TA). Write down the
differential equation for dT/dt
4. (20 pts) The golden toad of Costa Rica became extinct in 1989; the first
documented casualty of global warming. Let’s model the toad population in
the pond by differential equation P’ = .05P – 20 where starting population
is 300 toads and time t is measured in years. Solve the equation and
calculate how long it took for the toad to become extinct.
Extra Credit: In problem 2 assume logistic growth with carrying capacity of 2000.
Write down the logistic equation. ( 5 pts)