Math: Half-Life

Question

The Lualailua Hills Quadrangle of the East Maui volcano on the island of Maui in Hawaii is no loner active. To find out the date of the last eruption scientists conducted a chemical analysis of samples from the volcano area. The samples contained approx 62.31% of its original carbon-14. When, to the nearest year was the last eruption of the volcano? (use 5715 years for the half life of carbon-14)

Use: P=P₀e^rt

Answer 1

Set the exponential function up using P = 0.6231P₀. Then substitute the right hand expression for P in the exponential equation.

 

0.6231P₀ = P₀e^rt

 

Divide both sides by P₀ 

 

0.6231 = e^rt

 

So now the issue is that you do not know r or t. Use the half life given to work out what r is. Rather than memorize a formula, follow this procedure. After one half life, you have 50% of the original material. So, P = 0.5P₀. Same logic as used above.

 

0.5P₀ = P₀e^rt

 

Divide both sides by P0 and substitute in 5715 for t and solve for r.

 

0.5 = e^5715r

 

ln(0.5) = 5715r

 

r = ln(0.5) ÷ 5715 = -1.213 × 10^-4

 

Now substitute this value in for r and solve for t

 

0.6231 = e^rt

 

0.6231 = e^{(-1.213 × 10^-4)t}

 

ln (0.6231) = (-1.213 × 10^-4)t

 

t = ln(0.6231) ÷ (-1.213 × 10^-4) = 3900 years

 

 

 

 

Answer 2

Since the half-life is 5715, (1/2)P₀ = P₀e^5715r

 

                                           1/2 = e^5715r

 

                                           ln(1/2) = 5715r

 

                                                  r = -0.00012

 

So, P = P₀e^-0.00012t

 

Find t so that P = 0.6231P₀

 

      0.6231P₀ = P₀e^-0.00012t

 

        ln(0.6231) = -0.00012t

 

                    t = 3942

 

The volcano erupted 3942 years ago.  

 

                                                   

 

 

Answer 3

0.6231P_o = P_o(0.5)^t/5715

0.6231 = 0.5^t/5715

ln 0.6231 = (t/5715)(ln 0.5)

-0.4730 ≈ (t/5715)(-0.6931)

0.6824 ≈ t/5715

3,900 ≈ t