Relate Rates: Welfare cases depend on population

The number of welfare cases in a city of population p is expected to be W = 0.003p^4/3. If the population is growing by 700 people per year, find the rate at which the number of welfare cases will be increasing when the population is p = 1,000,000.

Hi Natalie, welcome to the Matheno community forum!

So, the number of welfare cases is given by:

We infer from the problem statement that W is a function of p and p is a function of t. Then we can apply the chain rule to W, taking the derivative with respect to time since the question asks us to find the rate at which W increases:


\dfrac{dp}{dt} is given as 700 people/year

You can find \dfrac{dW}{dp} by taking the derivative of W with respect to p, using the power rule.

Once you have these quantities, insert the given values p=1000000 and \dfrac{dp}{dt}=700. Be careful when writing down your final answer to give a whole unit answer (there are not 0.9 people.)

Please let us know if you need additional assistance!

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