Page 169 - How To Solve Word Problems In Calculus

To determine the time to reduce to 90 grams, we let y(t) = 90.

ln 2 t
90 = 100e − 5730

ln 2
− t
0.9 = e 5730

ln 2
− t = ln 0.9
5730
5730(ln 0.9)
t =−
ln 2
t ≈ 870.98

It will take approximately 871 years to decay to 90 mg.
14 C, a radioactive isotope of carbon, as well as 12 C, a stable
isotope, are both found in the atmosphere. Plants, which ab-
sorb carbon dioxide from the air, have the same ratio of 14 Cto
12 C as the atmosphere. The same is true for animals that eat
plants.
When a plant or an animal dies, it no longer absorbs car-
bon dioxide. Its 14 C decays but its 12 C does not. By measuring
the ratio of 14 Cto 12 C, scientists can approximate the age of
a fossil by comparing this number to the known ratio of 14 C
to 12 C in the atmosphere today. (The assumption is made that
this ratio has not changed throughout the years.)
This ratio decreases exponentially and has the same
half-life as 14 C, which is 5730 years.

EXAMPLE 4
14 C
How old is a fossil whose ratio is 10 percent of that found
12 C
in the atmosphere today?

Solution
Let r(t) represent the ratio of 14 Cto 12 C present after
t years. Since r(t) decreases exponentially,

r(t) = r 0 e − kt

156