need help with pre-calculus. really hard?

Question

The half-life of carbon-14 is 5730 years. How long does it take for 2.9 grams of carbon-14 to be reduced to 0.5 grams of carbon-14 by radioactive decay?

you have to use that formula.

A=A 0 e^rt

Answer 1

You use the formula A = A0*e*rt, but you don't immediately know r. But you know that the half-life, th, is the time it takes for the value of A to be reduced to .5A. At t=0, A = A0, at t=th, A = A0/2; from the second relation get

A0/2 = A0*e^r*th.

.5 = e^r*th; solve for r

ln(.5) = r*th; then r = ln(.5)/th; putting that back into the original formula gives

A=A0*e^[ln(.5)*t/th]

Since ln(.5) = - ln(2), this is also expressed as

A=A0*e^-[ln(2)*t/th]

At t = 0 A = 2.9 grams, so A0 = 2.9 grams

Af = .5 grams

Af = A0*e^-[ln(2)*tf/th]

Solve for tf as above: ln(Af/A0) = -ln(2)*tf/th