Answer to Question #17244 in Calculus for hsd

Question

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

2x^2+1xp+50p^2=26000.

If the price is increasing at a rate of 2 dollars per month when the price is 20 dollars, find the rate of change of the demand.

Rate of change of demand =

Answer 1

p=20 so 2x^2+x*20+50*20^2=26000
x=-60 or x=50
but x>0 so x=50
x, p functions of time t
and we know that
dp/dt=2 at point x=50 p=20
we must find dx/dt
differentiate equation
4xdx+x*dp+p*dx+100*p*dp=0 (: dt)
4x*dx/dt +x*dp/dt+100*p*dp/dt=0
for dx/dt we have
4*50*dx/dt+50*2+100*20*2=0
2*dx/dt+1+20*2=0
dx/dt=-41/2
rate of change of the demand=-41/2