Half life of radium (PRECALCULUS HELP PLEASE)

Question

The half life of radium is approximately 1690 years. If a lab has 50mg of Radium, find how long it takes for the substance to decay to 40mg, to the nearest year.

Answer 1

For these types of problems I find it easiest to use the following equation:

fraction remaining = 0.5ⁿ where n is the number of half lives that have elapsed.

So here we are looking for the number of half lives.  We can calculate the fraction remaining from the given info.

fraction remaining =40 mg/50 mg = 0.8

Substituting this in the equation we have 0.8 = 0.5ⁿ

log 0.8 = n log 0.5

-0.0969 = -.301 n

n = 0.322 half lives have elapsed

Since 1 half life is 1690 years, 0.322 half lives will be 0.322 x 1690 = 544 years