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 y cents per gallon, then approximately H hybrid cars will be sold each year, where

H(x,y)=7000-20x^(1/2)+8(0.1y+18)^(3/2)

She estimates that t years from now, the hybrid cars will be selling for 35000+350t dollars apiece and that gasoline will cost 300+18t^(1/2)cents per gallon. At what rate will the annual demand for hybrid cars be changing with respect to time 2 years from now?

Rate: cars per year (each year)

Explanation / Answer

x = 35000 + 350 t

y = 300 + 18 sqrt(t)

H(x,y) = 7000 - 20 sqrt(x) + 8 (0.1 y + 18)^1.5

= 7000 - 20 (35000 + 350 t)^0.5 + 8 ( 0.1 * (300 + 18 sqrt(t)) + 18)^1.5

= 7000 - 20 (35000 + 350 t)^0.5 + 8 ( 48 + 1.8 t^0.5)^1.5

hence

dH/dt = -[20*0.5*350]/(35000+350t)^0.5 + [8*1.5*(48+1.8t^0.5)^0.5*1.8*0.5]/t^0.5

thus rate in 2 years is ( t = 2)

dH/dt = - 18.52 + 54.29

= 35.77

hence rate ~ 36 cars

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