Chapter 5.1, Problem 35E

Radioactive Decay A sample of $8$ grams of radioactive material is placed in a vault. Let $P\left(t\right)$ be the amount remaining after $t$ years, and let $P\left(t\right)$ satisfy the differential equation $P\text{'}\left(t\right)=-0.021P\left(t\right)$.

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

b. What is $P\left(0\right)$?

c. What is the decay constant?

d. How much of the material will remain after $10$ years?

e. Use the differential equation to determine how fast the sample is disintegrating when just $1$ gram remains.

f. What amount of radioactive material remains when it is disintegrating at the rate of $0.105$ gram per year?

g. The radioactive material has a half-life of $33$ years. How much will remain after years? $66$ years? $99$ years?