Science:Math Exam Resources/Courses/MATH102/December 2017/Question 12 (a)/Solution 1

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 ${\textstyle k>0}$.