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.)

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.

To find the prices that will generate the maximum income from ticket sales, we need to maximize the revenue function.

First, let's write down the revenue function. The revenue is calculated by multiplying the number of adult tickets sold by the price of the adult ticket, and multiplying the number of child tickets sold by the price of the child ticket.

Revenue = N * A + M * C

Substituting the given equations for N and M:

Revenue = (80,000 - 2000A) * A + (30,000 - 1500C - 500A) * C

Now, we need to take the partial derivatives with respect to both A and C to find critical points:

d(Revenue)/dA = 80,000 - 4000A - 500C - 500A = -4500A - 500C + 80,000
d(Revenue)/dC = -1500A + 30,000 - 1000C

We set both partial derivatives equal to zero to find the critical points:

-4500A - 500C + 80,000 = 0
-1500A + 30,000 - 1000C = 0

Now we solve these equations simultaneously. The solution will give us the values of A and C that maximize the revenue.