Exponential decay given half life...? Should be easy..?

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Marmalade16

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Hello, I'm stuck on the second part of a simple exponential decay question. I solved part a:
a) 'A 50mg sample of cobalt-60 decays to 40mg after 1.6 minutes. Determine the half life of cobalt-60. ---I've correctly determined the half life to be about 5 minutes.

I know it should be simple, but for some reason I can't figure out part b, can anyone show me how to do it? I'd appreciate it!

b) How long will it take for the sample to decay to 5% its initial amount?
By the way, the textbook says the answer should be approx. 21.6 mins.
 
5% of the initial amount of 50 mg is 0.05*50 = 2.5 mg. So find the time it takes for the 50 mg to decay to 2.5 mg by using the half-life formula:

Y = A(0.5)^(t/h) , where Y is the ending amount, A is the starting amount, and t is the time, and h is the half-life; you are solving for t

2.5 = 50(0.5)^(t/5) (divide both sides by 50)
0.05 = 0.5^(t/5) (take log base 10 of both sides)
log 0.05 = log (0.5^(t/5)) (exponent property of logs)
log 0.05 = (t/5)log 0.5 (divide both sides by log 0.5)
(log 0.05)/(log 0.5) = t/5 (multiply both sides by 5)
(5log 0.05)/(log 0.5) = t
21.6096405 = t
t = about 21.6 minutes <===ANSWER
 
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