6. a. At a price of $8 per ticket, a musical theatre group can fill every seat i
ID: 1129816 • Letter: 6
Question
6. a. At a price of $8 per ticket, a musical theatre group can fill every seat in the theatre, which has a capacity of 1500. For every additional dollar charged, the number of people buying the ticket decreases by 75. Use calculus to find what ticket price maximizes revenue. Be sure to check your second order condition. b. Suppose that the number of people coming to a concert was N- P3 +2P, where N is the attendance and P is the price. Now suppose that price is dependent on the cost of putting the concert together P 5-C where C is the cost. Find dN/dC by the chain rule (please note that this question is separate from part a.).Explanation / Answer
Let N be the number of seats in the theatre.
It is given in the question that for every additional dollar charged, N decreases by 75.
At p=$8, N=1500
Revenue= PN= 8*1500= $12000
At p=$9, N=1425 (1500-75)
Revenue= 9*1425= $12,825
At p=10, N=1350 (1425-75)
Revenue= 10*1350= $13500
At P=$11, N=1275 (1350-75)
Revenue= 11*1275= $14,025
At P=$12 , N=1200 (1275-75)
Revenue= 12*1200= $14400
At P=$13, N=1125 (1200-75)
Revenue= 13*1125= $14,625
At p=14, N= 1050 (1125-75)
Revenue= 14*1050= $14,700
At P=15, N=975 (1050-75)
Revenue= 15*975 = $14,625
At p=16, N=900 (975-75)
Revenue= 16*900 = 14,400
So revenue will continue to decrease as price rises.
So, revenue is maximized when ticket price= $15.
2) N=P3 +2P and P=5-C2
Put values of P in equation of N=p3 +2p
So, N= (5-C2 )3 +2(5-C2 )
Differentiate with respect to C.
dN/dC = 3(5-C2)2 (-2C) -4C
= -6C3 +60C2 -4C-150.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.