A sorority plans a bus trip to the Great Mall of America during Thanksgiving bre

Question

A sorority plans a bus trip to the Great Mall of America during Thanksgiving break. The bus they charter seats 44 and charges a flat rate of $350 plus $35 per person. However, for every empty seat, the charge per person is increased by $2. There is a minimum of 10 passengers. The sorority leadership decides that each person going on the trip will pay $35. The sorority itself will play the flat rate and the additional amount above $35 per person.
a. For what number of passengers will the bus company's revenue be greatest? For what number of passengers will the bus company's revenue be least?
b. For what number of passengers will the amount the sorority pays be greatest? For what number of passengers will the amount the sorority pays be least?

Explanation / Answer

C(x)= cost to charter-ers for x people C(x)=350+bx, with b the rate per person. b(x)=35+2*(44-x), to reflect the original rate and also the empty seats. C(x)=350+x*(35+88-2x) Also, the cost to the charter-ers is the same as the revenue of the bus company. C(x)=R(x) a. R(x)=350+x*(123-2x) R'(x)=x(-2)+(123-2x)=123-4x=0, x=30.75, so we ought to check 30 and 31 since those are our options. R(30)=2240 R(31)=2241 (since 30 is farther from 30.75 than 31 is, 31 is closest to the peak) Greatest revenue: 31 Least revenue: 0 (only $350 for the booking) b. Let S(x) be the cost to the sorority. The sorority has to take on the entire cost, C(x) but the girls pay 35 per person to defray the cost. S(x)=C(x)-35x S(x)=350+x*(123-2x)-35x S'(x)=123-4x-35=0, x=22 Greatest cost: 22 Least cost: 0 and 44 (since at 0, they pay the booking, and at 44, the girls pay for their trips and the sorority only pays for booking)

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