A perfectly competitive firm produces two goods, X and Y, which are sold at $54 and $52 per unit, respectively. The firm has a total cost...

The total cost is given by TC = 3x^2 + 3xy + 2y^2 -
100.

The total revenue is: TR = 54x +
52y

The profits made are P = TR - TC = 54x + 52y - 3x^2 -
3xy - 2y^2 + 100.

As x + y = 10, we can replace y with 10 -
x. This gives:

P = 54x + 52(10 - x) - 3x^2 - 3x(10 - x) -
2(10 - x)^2 + 100

=> 54x + 520 - 52x - 3x^2 - 30x +
3x^2 - 200 - 2x^2 + 40x + 100

=> P = 12x - 2x^2 +
420

To maximize we solve dP/dx =
0

=> -4x + 12 =
0

=> x = 12/4 = 3

y =
10 - x = 7

To maximize profits 3 of x and 7
of y should be produced.