Question

A price p (in dollars) and demand x (in items) for a product are related by2x^2+2xp+50p^2=6200If price is increasing at a rate of $2 per month when the price is $10. Find the rate of change of the demand with respect to time.

Answer 1

A price p (in dollars) and demand x (in items) for a product are related by

`2x^2+2xp+50p^2=6200`

If price is increasing at a rate of $2 per month. Therefore,

`\frac{dp}{d\text{ t}}=2`

Now, from the given equation using p=10,

`\begin{gathered} x^2+10x+2500^{}=3100 \\ x^2+10x-600=0 \\ (x+30)(x-20)=0 \\ x=-30,20 \end{gathered}`

Since, demand can't be negative, therefore,

`x=20`

Now, differentiating the given equation w.r.t t and then putting x=20,p=-10,

`\begin{gathered} 4xx^(\prime)+2xp^(\prime)+2px^(\prime)+100pp^(\prime)=0 \\ 4(20)x^(\prime)+2(20)(2)+2(10)x^(\prime)+100(10)(2)=0 \\ 80x^(\prime)+80+20x^(\prime)+1000=0 \\ 100x^(\prime)=-1040 \\ x^(\prime)=-10.40 \end{gathered}`