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 dollars apiece and the price of gasoline is cents per gallon, then approximately hybrid cars will be sold each year, where H(X, Y) =7000-16x^(1/2)+4(0.1y+14)^(3/2) She estimates that years from now, the hybrid cars will be selling for 34000+300t dollars apiece and that gasoline will cost 300+10t^(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 = 34000 + 300t

y = 300 + 10t^(1/2)

hence, dx / dt = 300

and, dy /dt = 10*(1/2)*t^(-1/2) = 5(t^(-1/2))

H =7000-16x^(1/2)+4(0.1y+14)^(3/2)

Hence, del H / del x = -16*(1/2)*x^(-1/2) = -8*x^(-1/2)

and del H / del y = 4*(3/2)*(0.1y+14)^(1/2) *0.1 = 0.6*(0.1y+14)^(1/2)

dH / dt = (del H / dx)*(dx / dt) + (del H / del y)*(dy / dt)

dH / dt =-8*x^(-1/2) *300 +0.6*(0.1y+14)^(1/2)*5(t^(-1/2))

dH / dt =-2400*x^(-1/2) + 3*(0.1y+14)^(1/2)*(t^(-1/2))

dH / dt = -2400*(34000 + 300t)^(-1/2) + 3[(0.1(300 + 10t^(1/2))+14]^(1/2)*(t^(-1/2))

dH / dt = -2400*(34000 + 300t)^(-1/2) + 3[44+ t^(1/2)]^(1/2)*(t^(-1/2))

At t = 2, we get

dH / dt =-2400*(34000 + 300*2)^(-1/2) + 3[44+ 2^(1/2)]^(1/2)*(2^(-1/2))

dH / dt = -12 + 14.295

dH / dt = 2.295 cars per year

(This is correct answer. Above answer is wrong.)

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