Half life / Decay question

Question

I have 10 pounds of radioactive material buried in my backyard. The half-life of the material is 2345 years.

 

a.) How long will it take for 5 pounds of the material to decay?

b.) How long will it take for there to be 2 pounds remaining?

c.) How much is remaining after 1000 years?

Answer 1

PART A: For this part of the question you need to basically find the time it takes to get 5 lbs of material left when you start out with 10 lbs having a half-life of 2345 years.

 

Step#1 use the formula A= A₀ × t/h  where A₀ =10 and t=? (Unknown) h=2345 and A=5

5= 10 × (1/2)^t/2345  start by moving the 10 to the other side multiply both sides by 1/10

1/2 = (1/2)^t/2345  next we take the log of both sides

log(1/2)= log( (1/2)^t/2345 )   then we bring down the exponent

log (1/2)= t/2345 × log(1/2)  next we multiply both sides by 2345

2345 × log (1/2) = t × log(1/2) next multiply both side by 1/log(1/2)

( 2345 × log (1/2) ) / log(1/2) = t  now solve for t you will get t=2345

PART B: same as part A just need to use A=2 whereas in part A we used A=10

PART C: using the half-life formula A= A₀ × t/h  we simply plug in the values

A= 10 × (1/2)^1000/2345 => 10  × (1/2) ^.4264 => 10  × 0.7441 => 7.44 lbs would be left after 1000 years. 

 

Great problem. I hope I was detailed enough for you to follow through.