Question

Hi there I need help on my calculus homework and this week, we were working on partial derivative. I don't know how to approach this word problem and set it into a maximum or minimum. Well I am not too sure if this problem is a minimum or maximum problem. Please help me and this is the homework problem below

A football stadium has a capacity of 50,000. The owners calculate that for an average game, if they charge $A for an adult ticket and $C for a child ticket, then N adults and M children will try to obtain tickets, where:
N = 80,000 - 2000A and
M = 30,000 - 1500C - 500A
Find the prices that the owners should charge in order to generate the maximum income from ticket sales for a game. (NOTE: It may not be necessary for the stadium to be full to maximum income.)

Answer 1

Income=A*N + C*M
=A(80000-2000A)+C(30000-1500C-500A)
multiply it out
Take derivitive with respect to A or C, I will do C first:

dI/dc=30000-3000C set it to zero, then
C=10

Now, do dI/dA
80000-4000A -500C=0
or A=80000-5000)/4000=75/4=17 dollars

Check: make certain the stadium can hold N +M
N=46000 M=6500 so yes it can.

check this.