Science:Math Exam Resources/Courses/MATH102/December 2012/Question B 05

MATH102 December 2012

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Question B 05

In the Chernobyl reactor explosion, which occurred on April 26, 1986, substantial amounts of the isotope strontium-90 (90Sr) contaminated the area around the nuclear plant. 90Sr decays at a rate proportional to its quantity. 90Sr has a half-life of 29 years; that is, it takes 29 years for a quantity of 90Sr to decrease by half. What is the proportion of 90Sr originally released which remains on April 26, 2012?

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, described by the formula $\displaystyle y(t)=Ce^{kt}$, where $y(t)$ is the percentage of material left after t years, and C is the initial amount.

Hint 2
Before you can calculate the current amount remaining, you will have to use the information about half-life to find k.

Hint 3
With a half life of 29 year, expect more than 50% of the material to be remaining at 26 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. As stated in the second hint, we will first use the information about half-life to solve for the rate of decay, or k in the equation: $\displaystyle y(t)=Ce^{kt}$ If $y(t)$ is the percentage remaining after $t$ years, we know that $50\%$ of 90Sr remains after 29 years. At t = 0 the amount is 100 percent, hence choose C = 100. Plugging this into the formula gives: $\displaystyle 50=100e^{k29}$ Solving for $k$, we get $\ln({\frac {1}{2}})=29k$ $\displaystyle k=\ln(1/2)/29$ Plugging this back into our formula for $y(t)$, we can now solve for the percentage remaining on April 26, 2012, when $t=26$. $\displaystyle y(t)=100e^{(\ln(1/2)/29)*26}$ Simplifying will give: $y(t)=100*({\frac {1}{2}})^{26/29}$ or 53% remaining.

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