Page 178 - How To Solve Word Problems In Calculus

(a) How many people initially contracted the disease and how many
people contracted the disease within the next two weeks?
(b) At what rate did people contract the disease after two weeks?
(c) When did the rate of infection begin to decline?
(d) If left untreated, how many people would eventually contract
the flu?
◦
11. On a day when the temperature is 30 Celsius, a cool drink is taken
◦
from a refrigerator whose temperature is 5 . If the temperature of
the drink is 20 after 10 minutes, what will its temperature be after
◦
20 minutes?
Solutions to Supplementary Problems

1. Let y(t) represent the size of the population after t hours. Since the
rate of growth of the bacteria is proportional to the number of
dy
kt
bacteria present, = ky and it follows that y = y 0 e . When
dt
t = 0, y 0 = 100 and the population function becomes
kt
y(t) = 100e . Since the population doubles every 2 hours, y = 200
when t = 2. This allows us to determine k.
y = 100e kt

200 = 100e 2k

2 = e 2k
2k = ln 2

1
k = ln 2
2
We may now write the complete population function:

1 ln 2)t
y(t) = 100e ( 2
t ln 2
= 100e 2

3 ln 2
(a) y(3) = 100e 2 ≈ 100(2.828) = 282.8. There are 283 bacteria
after 2 hours.
(b) We wish to determine the value of t for which y(t) = 5000.

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