The effectiveness of a police force may be measured by its clearance rate: the number of charges laid in a month divided by the total number of unsolved crimes.

In Arachnid Boy's home town, new crimes are reported roughly $20$ times per month, and while Arachnid boy is in town, the police clearance rate is $40\%$. Arachnid Boy comes back from his holiday and finds there are $100$ unsolved crimes.

Let $x$ be the number of unsolved crimes at the start of month $t$ , with $t=0$ representing the first month that Arachnid Boy is back from his holiday. What is the value of $\dfrac{dx}{dt}$?

All I've gotten to is

$$\dfrac{dx}{dt} = 100 - 8x$$

although I know that this expression is incorrect. I've tried to consider a linear relationship and solving for constants using the initial conditions, however I highly doubt this is the correct way to attempt this question.

Any help or guidance is greatly appreciated!


Each month, $20$ new cases come in. That means $\frac{dx}{dt}$ gets a $+20$ contribution from that. Also, each month, they clear $40\%$ of all cases, so $\frac{dx}{dt}$ gets a $-0.4x$ contribution from that. These are the only pieces of information that we are given regarding how the number of cases changes from month to month.

Which is to say, $$ \frac{dx}{dt}=20-0.4x $$ The $100$ cases is an initial value, and doesn't affect this expression for $\frac{dx}{dt}$ at all. Of course, if you want to solve this initial value problem, the $100$ must be used.

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