Radium decays exponentially. Its half-life is 1,690 years. How long will it take for a 50-gram sample of radium to be reduced to 5 grams?

Question

1/2P = Pe^(-1690k)

(ln1/2)/(-1690) = k

4 x 10^-4 = k

5 = 50e^-kt

ln0.1 = -kt

ln0.1/-k = t

5,630 years = t

...but textbook answer is 5,614 years. Where did I go wrong?

Answer 1

I got 5616 using 4.1 * 10^-4 for k

You did it right but did not carry the same number of significant figures.

it is easier to use a^-b = 1/a^b in a case like this
1/10 = e^-kt
becomes
e^kt = 10
t = ln 10/k = 2.303/.00041

Answer 2

your calculator. Set it on scientific, so the k is equal to 4.101E-4

With that, I get 5615 years.

Answer 3

I, too, get 5615.