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

MATH102 December 2017
Other MATH102 Exams

### 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!

 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. Math Learning Centre A space to study math together. Free math graduate and undergraduate TA support. Mon - Fri: 11 am - 5 pm, Private tutor We can help to