Page 166 - How To Solve Word Problems In Calculus

Thus

1
( ln 2)t
y(t) = 500e 4
t ln 2
= 500e 4
= 500(e ln 2 ) 4 t
t
= 500(2) 4

To determine the population size after 6 hours (t = 6), we com-
pute y(6) = 500(2) 1.5 ≈ 1414.
Exponential decay occurs when a substance deteriorates
with time. If the rate of decay is proportional to the amount
of substance present, its mass satisfies the equation

dy
=−ky
dt
In this equation, k is a positive constant. The minus sign pre-
ceding it indicates that the rate of change is negative, i.e., the
mass is decreasing. Solving this equation in the manner de-
scribed above, we are led to the solution y = y 0 e −kt .

EXAMPLE 2
A radioactive substance has a mass of 100 mg. After 10 years
it has decayed to a mass of 75 mg. What will the mass of the
substance be after another 10 years?

Solution
Let y(t) represent the mass of the substance after t years.
We know y(0) = 100 and y(10) = 75. We wish to determine
the value of y(20).

y(t) = y 0 e − kt
y(t) = 100e − kt

First we need to determine k. This can be done by substituting
t = 10.

153