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Five math problems from a midterm exam for math 30. The problems involve calculating the percentage of customers getting a free drink based on waiting time, finding a function passing through a point with a given slope, plotting fish population over time, and writing but not solving differential equations for virus infection and radioactive decay.
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Midterm 3, Math 30, April 28, 2008
time is^ μ μ
t p t e
1 ( ) where μ is the average time.) (10 pts)
Find the function f(x) that passes through a point (1,-1) and whose slope at (x,y) is y 2 x. (10 pts)
Trout population in the lake is modeled by the equation.
P^2 ( P 1500 )( 2000 P ) dt
dP (^) = − −
a) Plot the fish population over time if the starting population is 1000. (5 pts) b) Plot the fish population over time if the starting population is 2100. (5 pts) (Note: In your drawing clearly label the numbers for the starting and final populations after a long time)
a) A virus infects cells at the rate proportional to the product of the fraction of the infected cells and the fraction of healthy cells. Let I(t) be the fraction of infected cells. (5 pts)
b) Radioactive atoms decay at the rate proportional to their number. Let N(t) be the number of radioactive atoms. (5 pts)
dP = λ. If λ = .01/hour, how long will it
take for the population to double? (10 pts)