Given a radioactive substance, experiments have found that its decay rate is a c

Question

Given a radioactive substance, experiments have found that its decay rate is a constant proportion of the amount of the substance. Suppose we have 100 grams of radioactive Carbon-14, ^14 C. Write down a differential equation initial value problem that models the mass of^14 C after t years. The half-life of^14 C is about 5730 years. Find the mass that remains after t years by solving the initial value problem in part (a). When a plant or animal dies, it stops replacing its carbon and the amount of^14 C begins to decrease through radioactive decay. Suppose an archaeological discovery revealed a papyrus fragment that had about 70% as much^14 C as does the plant material on the earth today. Estimate the age of the papyrus.

Explanation / Answer

a)If the amount = 95
95 = 100 e^(1k)
.95 = e^k
k = ln .95

for half-life time, only 50 g of the original 100g would remain
50 = 100 e^(ln.95 t)
.5 = e^(ln.95 t)
ln.95t = ln.5
t = ln.5/ln.95 = appr13.5 years

b,so A = 100 e^(ln.95 t)
when t = 50
A = 100 e^(50ln.95) = 7.69 g are left

c,
0.7 = (1/2)^(t / 5730)
ln(0.7) = (t / 5730) * ln(0.5)
5730 * ln(0.7) / ln(0.5) = t
5730 * 0.5145 = t
2948.504= t

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