How do I exactly solve this word Problem?

Question

A zircon (a mineral common in igneous rocks) sample contains 3,500 atoms of radium. If the half-life of radium is 1,620 years, how many atoms will be present after 345 years?

It's from my Calculus Class, I need solutions to compare to mine, thanks!

Answer 1

345 is far less than 1620, so you know far more than 1/2 of 3500 will be left. Whatever you get, you know it's far more than 1750 atoms. It's between 1750 and 3500. the average plus or minus 500 would be in the right range.

just a check on whether your answer makes sense

1/2 = e^-rt = e^-1620r

take logs

ln(1/2) = -1620r

solve for r

r = ln(1/2)/-1620 = about 4.28 x 10^-4 = 0.000428

for 345 years

A = 3500e^-000428(345) = 3500e^-.1476 = 3500(1/1.159049184) = 3500/1.159049184 = about 3019.7 atoms left

A=3019.7 atoms

or

you could also calculate it

as A=3500(1/2)^(345/1620) which also = 3019.7 atoms rounded off to nearest decimal

Answer 2

A = 3500(0.5)^345/1620

Can you calculate and answer?

Look up the formula for half-life.