Science:Math Exam Resources/Courses/MATH100/December 2012/Question 05 (a)

MATH100 December 2012

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Question 05 (a)

Full-Solution Problems. In questions 5-11, justify your answers and show all your work. If a box is provided, write your final answer there. Unless otherwise indicated, simplification of numerical answers is required in these questions. 5. A sample of the radioactive substance Rhodium-101 decayed to 12.5% of its original amount after 10 years. (a) What percentage of the original amount is remaining after another 5 years?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Recall that radioactive decay is an example of exponential decay. So use the formula $\displaystyle y(t)=Ce^{kt}$, where $y(t)$ is the percentage of material left after t years.

Hint 2
Now use the data in the question to solve for C (original amount), then k (rate of decay), and finally y(15) (five years after the first 10 years).

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Proceeding as the hint, let $\displaystyle y(t)=Ce^{kt}$ be the percentage of material remaining after t years. At time 0, we have 100% left, so we write: $\displaystyle {\begin{aligned}100&=y(0)\\&=Ce^{k0}\\&=C\end{aligned}}$ Next, we know that after 10 years, there is 12.5% left, thus using $C=100$: $\displaystyle {\begin{aligned}12.5&=y(10)\\&=100e^{10k}\end{aligned}}$ Dividing by 100 and taking natural logarithms of both sides will give: $\displaystyle \ln(12.5/100)=\ln(e^{10k})$ Because $e^{x}$ and $\ln(x)$ are inverses, the right side simplifies to: $\displaystyle \ln(12.5/100)=10k$ And we can solve for k: $\displaystyle k=\ln(12.5/100)/10=\ln(1/8)/10$. Now, we are looking for the percentage remaining after 15 years (5 years after the initial 10 years), or $\displaystyle y(15)$, so plugging in the data gives $\displaystyle y(15)=100e^{15(\ln(1/8)/10)}=100e^{\ln((1/8)^{15/10})}=100(1/8)^{3/2}$ completing the question.

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