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

MATH102 December 2017

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Question 12 (b)

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. (b) Suppose telescope images from $100$ years ago reveal that four fifths of the area of Dione (a moon of Saturn) was free of craters (i.e. $A(0)=4/5$) and that recent images show that this fraction has decreased by $10^{-7}$ to $A(100)=4/5-10^{-7}$. Show that the value of the constant of proportionality in your equation from (a) is given by $k=-{\frac {1}{100}}\ln \left(1-{\frac {5}{4}}10^{-7}\right)$ by solving the differential equation and using the information provided by the telescope imaging.

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
Recall that the solution of the differential equation in part (a) is $$A(t)=A(0)e^{-kt}.$$

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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. Solving the differential equation in part (a), we have, as in the Hint, $$A(t)=A(0)e^{-kt}.$$ We are given that ${\textstyle A(0)={\dfrac {4}{5}}}$ and ${\textstyle A(100)={\dfrac {4}{5}}-10^{-7}}$. Therefore, we have $$A(t)={\dfrac {4}{5}}e^{-kt},$$ and, putting ${\textstyle t=100}$, $${\dfrac {4}{5}}-10^{-7}={\dfrac {4}{5}}e^{-100k}.$$ We solve this equation for ${\textstyle k}$: $$1-{\dfrac {5}{4}}10^{-7}=e^{-100k}$$ $$-100k=\ln \left(1-{\dfrac {5}{4}}10^{-7}\right)$$ $$k=-{\dfrac {1}{100}}\ln \left(1-{\dfrac {5}{4}}10^{-7}\right).$$