# 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 function given by TC = 3x^2+3xy+2y^2-100 Find the...

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 function given by TC = 3x^2+3xy+2y^2-100

Find the quantities of each good which must be produced and sold in order to maximise profits.

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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.**

Thanks justaguide for you help, but I realised I should have put a part (a) and part (b) on that question. (a) find the quantities of each good which must be produced and sold in order to maximise profits. (b) was the requirement that x + y = 10.

I checked the answers and that was the answer for part (b) x = 3 and y = 7. The answers for part (a) with maximising profits were x = 4 and y = 10, they aren't subject to a requirement, I'm not sure how to do the working for it. Thanks.