Math 181 - Quiz 5B, other situations with
difference equations - solutions
Problem 1 At noon, a boy has 800 milligrams of a certain drug in his blood.
Every hour, his kidneys remove the drug from 250 cc of blood, with a total
amount of 2000 cc of blood in his body. Write D(t) for the amount of the
drug in his blood at time t.
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 5pm?
d) At what time will there be only 20 milligrams of the drug left in the boy’s
blood?
Solution. a) First, r = −250/2000 = −0.125 so D(t + 1) − D(t) =
−0.125D(t).
b) D(t) = 800(1 − 0.125)t = 800(0.875)t if we put time t = 0 at noon.
c) Substitute t = 5 into b), so
D(4) = 800(0.875)5 ≈ 410.327
d) Solve D(t) = 20:
20 = 800(0.875)t
1
= 0.875t
40
− ln 40 = t ln(0.875)
ln 40
t = −
≈ 27.626
ln 0.875)