Science:Math Exam Resources/Courses/MATH102/December 2017/Question 12 (d)

MATH102 December 2017

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Question 12 (d)

The age of a solid planet can be estimated by calculating the fraction of the surface of the planet that is free of meteor impact craters. Assume that the rate at which the crater-free area decreases is proportional to that area itself. (d) Approximately how old is Dione? You may assume that Dione had no craters on its surface when it first formed. Your answer here should also be a simplified rational number (use the same linear approximation again). Remark: you can earn partial credit for this part by expressing the age in terms of $k$, even if you did not find the rational approximation in part (c).

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

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. 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. Let the age be ${\textstyle T}$. We need to solve the equation ${\textstyle A(-T)=1}$: $${\dfrac {4}{5}}e^{kT}=1$$ $$e^{kT}={\dfrac {5}{4}}$$ $$kT=\ln {\dfrac {5}{4}}$$ $$T={\dfrac {1}{k}}\ln {\dfrac {5}{4}}.$$ By the approximation in part (c), we have $$T\approx {\dfrac {4}{5}}\cdot 10^{9}\cdot \ln {\dfrac {5}{4}}.$$ By the linear approximation, $$\ln {\dfrac {5}{4}}\approx {\dfrac {5}{4}}-1={\dfrac {1}{4}}.$$ Therefore, $$T\approx {\dfrac {1}{5}}\cdot 10^{9}.$$ In other words, Dione is approximate ${\textstyle 2\cdot 10^{8}}$ years old. Answer: the age of Dione is approximately ${\textstyle {\color {blue}2\cdot 10^{8}}}$ years.