Sample Problems
Math 1100
13 November, 2013
1. When an organism dies, the amount of carbon-14 present decreases at a rate proportional
to the amount remaining. That is, it is modeled by the differential equation
dy
= ky
dt
where y is the amount of Carbon-14 remaining, k is some constant, and t is the time in
years. The half-life of Carbon-14 is 5730 years; that is, after 5730 years, half the initial
amount of carbon-14 remains.
Suppose scientists discover a fossil containing 1% of the original amount of carbon-14.
How old is the fossil?
2. When interest is compounded continuously, the rate of change of the amount of money
x in an account is proportional to the amount present. That is,
dx
= rt
dt
where r is the interest rate (a constant) and t is the time in years. If $10,000 is invested
at 6% interest, what will the value of the investment be after 5 years?
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3. A certain population of bacteria grows according to the differential equation
1 dy
·
=k
y dt
for some constant k. If the population doubles every two hours, how long will it take to
have 50 times the initial number of bacteria?
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