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:

W=0.003p^{4/3}

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:

\frac{dW}{dt}=\frac{dW}{dp}\frac{dp}{dt}

\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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