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

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