Question #17244

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 =

Expert's answer

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

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

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