How many grams?

Question

the half life of radium is 1690 years. If 50 grams are present now, how much will be present in 270 years?

Answer 1

ok 

 

we will use the following equation to do an exponential decay

 

y = Ce^kt

 

So this is what we know.  

 

In 1690 years we will have half of the material left.  So say if we start with 50 grams of material...in 1690 years we will have 25g of material left.  

 

So this is what we do:

 

25 = 50 x e^k(1690)

 

You divide 25 by 50 and get 1/2 = e^1690k

 

Take a natural log of both sides

 

and get ln 1/2 = 1690 x k     e and ln cancel each other out.  

 

ln of 1/2 = -.693

 

so you get -.693 = 1690k  Divide -.693 by 1690 to get the value of k.  This is an exponential decay.  k = -4.1 * 10^-4

 

Now you plug this k value back into 

 

y = Ce^kt

 

So y = 50e^{-4.1*10-4 (270)}

 

 

I don't have a calculator in front of me but you can solve the rest by plugging it into your calculator to solve for y.  This is the amount of material you will have left.  

 

Answer 2

A = A₀(1/2)^t/h, where A₀ = starting amount, t = time in years, h = half-life

 

A = 50(1/2)^270/1690

A = 50(1/2)^{0.1597633136094675}    I just used 0.16

A ≈ 50(0.895)

A ≈ 44.75