Suppose that price, $p$ in dollars, and the number of items sold, $x,$ are related by: $p^2 - xp + x^2 = 175.$
If $p$ and $x$ are functions of time, how fast is $p$ changing with respect to time when $p=\$ 10$ and $x$ is increasing by $5$ units per day?
So far I’ve got: $$\frac{dp}{dt} = \frac{-2x+p}{2p-x}.$$
Not sure what to do after this? Help would be extremely appreciated :(
Let's see: \begin{align*} \frac{d}{dt}[\,p^2-xp+x^2&=175\,] \\ \underbrace{2p\dot{p}}_{\text{Chain}}-\underbrace{(x\dot{p}+\dot{x}p)}_{\text{Product}}+\underbrace{2x\dot{x}}_{\text{Chain}}&=0 \\ \dot{p}(2p-x)&=\dot{x}(p-2x) \\ \dot{p}&=\frac{\dot{x}(p-2x)}{2p-x}. \end{align*} The issue is that the problem statement doesn't give you the value of $x$, but of $\dot{x}.$ You can find $x$ by solving the original equation for it when you've plugged in $p=10.$ That is, you are solving $$100-10x+x^2=175,\qquad\text{or}\qquad x^2-10x-75=0.$$ The solutions are $x=-5, 15.$ Can you rule out one of these? Why?