log equations (thank you in advance)

Question

Tritium has a half-life of 12 years.

a. If an initial sample of 15 grams is present, find a function T(x) that represents the amount remaining, in grams, after x years.

T(x) = _________________________

b. Use your function to determine the amount of time it takes for the Tritium to decay to 5 grams.

Answer 1

The formula to use is

A=A_oe^kt

We can plug in the values we know and solve for k

1/2(15) = 15e^12k 15 cancels

1/2= e^12k Take the ln of both sides

ln(1/2) = ln(e^12k) Apply exponent property

ln(1/2) = 12k(ln e) ln e=1

ln(1/2) = 12k Divide both sides by 12

k=-0.0578

Plug this in the equation

A=15e^-0.0578t

For the next part, plug in 5 and solve for t using the same process as we used for k

5=15e^-0.0578t Divide both sides by 15

(1/3) = e^-0.0578t Take the ln of both sides

ln (1/3) = ln(e^-0.0578t) Apply exponent property

ln (1/3) = -0.0578t ln(e) ln e=1

ln (1/3) = -0.0578t Divide both sides by -0.0578

t=19 years