Question

Plutonium-210 has a half-life of 140 days. Use the formula , where ,  is the remaining mass,  is the original mass, and is the half-life, to determine how long it takes to reduce 300 milligrams of plutonium-210 to 200 milligrams. Arrange the steps in the right order to solve the problem.

Answer 1

Since this is exponential decay we can express it as:

f=ir^t, f=final amount, r=common ratio, t=time

If the half life is 140 days we can say:

a/2=ar^140

.5=r^140 

r=.5^(1/140)  now we can express our equation as:

f=i(.5^(1/140))^t which is equal to:

f=i(.5)^(t/140)  now we want to find the time necessary to reduce 300mg to 200mg so:

200=300(.5)^(t/140)  divide both sides by 300

2/3=.5^(t/140) taking the natural log of both sides

ln(2/3)=(t/140)ln.5  divide both sides by ln.5

ln(2/3)/ln.5=t/140  multiply both sides by 140

t=140ln(2/3)/ln.5

t≈81.89 days (to the nearest hundredth of a day)