Science:Math Exam Resources/Courses/MATH100/December 2012/Question 05 (a)
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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.
Recall that radioactive decay is an example of exponential decay. So use the formula , where is the percentage of material left after t years.
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).
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Proceeding as the hint, let be the percentage of material remaining after t years.
At time 0, we have 100% left, so we write:
Next, we know that after 10 years, there is 12.5% left, thus using :
Dividing by 100 and taking natural logarithms of both sides will give:
Because and are inverses, the right side simplifies to:
And we can solve for k:
Now, we are looking for the percentage remaining after 15 years (5 years after the initial 10 years), or , so plugging in the data gives
completing the question.