Chapter 5.1, Problem 36E

Radioactive Decay Radium $226$ is used in cancer radiotherapy. Let $P\left(t\right)$ be the number of grams of radium $226$ in a sample remaining after $t$ years, and let $P\left(t\right)$ satisfy the differential equation

$P\text{'}\left(t\right)=-0.00043P\left(t\right),P\left(0\right)=12$.

a. Find the formula for $P\left(t\right)$?

b. What was the initial amount?

c. What is the decay constant?

d. Approximately how much of the radium will remain after $943$ years?

e. How fast is the sample disintegrating when just $1$ gram remains? Use the differential equation.

f. What is the weight of the sample when it is disintegrating at the rate of $0.004$ gram per year?

g. The radioactive material has a half-life of about $1612$ years. How much will remain after $1612$ years? $3224$ years? $4836$ years?