Math 181 - Quiz 5A, other situations with
difference equations - solutions
Problem 1 At noon, a woman has 2400 milligrams of a certain drug in her
blood. Every hour, her kidneys remove 14% of the amount D(t) of the drug
in the blood.
a) Find a difference equation for D(t).
b) Tell D(t) for all times t.
c) How much of the drug will be left at 4pm?
d) At what time will there be only 100 milligrams of the drug left in the
woman’s blood?
Solution. a) D(t + 1) − D(t) = −0.14D(t).
b) D(t) = 2400(1 − 0.14)t = 2400(0.86)t if we put time t = 0 at noon.
c) Substitute t = 4 into b), so
D(4) = 2400(0.86)4 ≈ 1312.820
d) Solve D(t) = 100:
100 = 2400(0.86)t
1
= 0.86t
24
− ln 24 = t ln(0.86)
ln 24
≈ 21.071
t = −
ln 0.86)