A car dealer determines that if gasoline-electric hybrid automobiles are sold fo

Question

A car dealer determines that if gasoline-electric hybrid automobiles are sold for x dollars apiece and the price of gasoline is ycents per gallon, then approximately H hybrid cars will be sold each year, where

H(x,y)=800012x^(1/2)+2(0.1y+12)^(3/2)

She estimates that t years from now, the hybrid cars will be selling for 37000+300t dollars apiece and that gasoline will cost 275+20t^(1/2)cents per gallon. At what rate will the annual demand for hybrid cars be changing with respect to time 3 years from now?

Rate = _________________ cars per year (each year)

Explanation / Answer

H(x,y)=800012x^(1/2)+2(0.1y+12)^(3/2)

H(t)=800012(37000+300t)^(1/2) +2(0.1*(275+20t^(1/2))+12)^(3/2)

H(t)=800012(37000+300t)^(1/2) +2*(39.5+2t^(1/2) )^(3/2)

dH/dt=0-12*(1/2)*300*(37000+300t)^(-1/2) +2*(3/2)*2*(39.5+2t^(1/2) )^(1/2)

dH/dt=[-1800*(37000+300t)^(-1/2)] +[6*(39.5+2t^(1/2) )^(1/2)]

dH/dt at t=3

=[-1800*(37000+900)^(-1/2)] +[6*(39.5+2*3^(1/2) )^(1/2)]

=30.1

Rate = 30.1 cars per year

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