Question

When the price of a certain commodity is p dollars per unit, the manufacturer is willing to supply x thousand units, where 𝑥 2 − 2𝑥√𝑃 − 𝑃 3 = 31 How fast is the supply changing the price when the price is $ 9 per unit and increasing at the rate of 20 cents per week 

Answer 1

$x^2-2x\sqrt{p}-p^3=31\\\frac{dp}{dt}=0.2\\p=9\Rightarrow x^2-6x-760=0\Rightarrow \left[ x\geqslant 0 \right] \Rightarrow x=3+\sqrt{769}\\2x\frac{dx}{dt}-2\frac{dx}{dt}\sqrt{p}-\frac{x}{\sqrt{p}}\frac{dp}{dt}-3p^2\frac{dp}{dt}=0\\\frac{dx}{dt}=\frac{\left( \frac{x}{\sqrt{p}}+3p^2 \right)}{2\left( x-\sqrt{p} \right)}\frac{dp}{dt}=\frac{\left( \frac{3+\sqrt{769}}{\sqrt{9}}+3\cdot 9^2 \right)}{2\left( 3+\sqrt{769}-\sqrt{9} \right)}\cdot 0.2=0.91322\,\,thousands\,\,units\,\,per\,\,week$