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May 1st, 2017, 06:46 PM   #1
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What did I do wrong in my solution?

A certain radioactive substance has a half-life period of 35 days. How long will it take a 2 gram sample to decay to 0.05 grams?

my solution

dy/dt = ky where y is the amount of the substance

Solving it gives lny=kt+c ------ (1)

When t=0 days, y= 2 grams. So, plugging them into (1) above gives

c=ln2

lny = kt + ln2

When t=35 grams, y=1 day. Next, I use the values to find k and get

k=-ln2/35

So, lny= - (ln2)t/35 +ln2 ------ (2)

When y = 0.05 grams, I plug it into (2). t=186 days.

This is wrong. The correct answer is 167 days.

I cannot get the answer. Can someone help me? Thanks.
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May 1st, 2017, 07:43 PM   #2
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I find $k=-\dfrac{\ln(2)}{35}$ just as you do and

I get 186.267 days for 2g to decay to 0.05g.

I think your answer key is mistaken.
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June 28th, 2017, 09:01 AM   #3
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Given that this substance "has a half life of 35 days", we can write $\displaystyle y(t)= A(1/2)^{t/35}$ where A is the original amount and t is in days. The original amount was 2 grams so $\displaystyle y(t)= 2(1/2)^{t/35}a$. To determine when it is reduced to 0.05 grams, solve $\displaystyle 2(1/2)^{t/35}= 0.05$.

Divide both sides by 2: $\displaystyle (1/2)^{t/35}= 0.025$.

Take the logarithm, base 10, $\displaystyle (t/35)log(1/2)= -(t/35)log(2)= log(0.025)=-1.602$. Since log(2)= 0.3010, t/35= 1.602/.3010= 5.322 and t= 35(5.322)= 186.2 days.
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