Page 165 - How To Solve Word Problems In Calculus

ln y = kt + ln y 0
ln y − ln y 0 = kt

y
ln = kt
y 0
y
= e kt
y 0

y = y 0 e kt

If the value of y 0 is known and the value of k can be deter-
mined, we can represent the amount of the substance as a
function of time.

EXAMPLE 1

A bacteria culture has an initial population of 500. After
4 hours the population has grown to 1000. Assuming the
culture grows at a rate proportional to the size of the popula-
tion, find a function representing the population size after t
hours and determine the size of the population after 6 hours.

Solution
Let y(t) represent the size of the bacteria population after
t hours. Since the rate of growth of the population is pro-
portional to the population size, it follows from the above
discussion that
y(t) = y 0 e kt

y(t) = 500e kt

Since y(4) = 1000, it follows that

1000 = 500e 4 k

2 = e 4 k
Recall that y = ln x ⇔ x = e y
4k = ln 2
1
k = ln 2
4
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