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

MATH102 December 2017

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Question 12 (a)
The age of a solid planet can be estimated by calculating the fraction $A(t)$ 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. (a) What differential equation does $A(t)$ solve?

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
Let $L$ be the the surface area of the planet. Then, the crater-free area is $A(t)L$.

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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. By the Hint, ${\textstyle A(t)L}$ is the crater-free area. Then, the rate at which the crater-free area decreases is $-(A(t)L)'=-A'(t)L$. Since it is proportional to that area itself, we have $$-A'(t)L=kA(t)L,$$ for some constant ${\textstyle k>0}$. By dividing both side of the equation by $L$, we have the equation for $A$, ${\textstyle {\color {blue}A'(t)=-kA(t)}}$, where ${\textstyle k>0}$.