Science:Math Exam Resources/Courses/MATH110/April 2011/Question 05
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Question 05 

In the 1986 Chernobyl reactor explosion, substantial amounts of the isotope strontium90 (^{90}Sr) contaminated the area around the nuclear plant. ^{90}Sr undergoes exponential radioactive decay with a halflife of 29 years. Write down a ``calculatorready`` expression for the percentage of the origina ^{90}Sr remaining today (use the year the exam was written). 
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

The formula for exponential radioactive decay is , where S_{0} is the initial quantity and k is a rate of decay. How can you use the information about the halflife of ^{90}Sr to find k? 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We know that the amount of ^{90}Sr remaining at a time is modeled by the equation , where is the initial quantity and k is a rate of decay. Our first task is to find k. Using the information that the halflife of ^{90}Sr is 29 years, we know that when t = 29, S(29) = S_{0}/2. Plugging this into our equation of exponential decay, we get: Note that k < 0 as expected: the amount of ^{90}Sr declines over time. Putting k back into the formula , we get You can leave this expression as is, or simplify it to get To find the percentage of ^{90}Sr remaining today, we simply plug in the amount of time that has elapsed since the initial explosion in 1986; in 2011, that is 25 years. So The percentage of remaining is thus 