# Math Help - Demand and total cost function

1. ## Demand and total cost function

I'm stuck on how to set this up, and help would be great.

Demand function is Q = 16 - P
Total cost function is defined as TC = 3 + Q + 0.25 Q^2.

Use the two functions to form the firm's profit function and then determine the level of output that yields the profit maximum. What is the level of profit at the optimum?

Do I put Q in the TC function to solve?
when I do this I get:

3+(16-p)+.25(16-p)^2
19+p+.25(256-32p+p^2)
19-p+64-8p+.25p^2
83-9p+.25p^2

If this is right where do I go next?

2. Originally Posted by rodimus
I'm stuck on how to set this up, and help would be great.

Demand function is Q = 16 - P
Total cost function is defined as TC = 3 + Q + 0.25 Q^2.

Use the two functions to form the firm's profit function and then determine the level of output that yields the profit maximum. What is the level of profit at the optimum?

Do I put Q in the TC function to solve?
when I do this I get:

3+(16-p)+.25(16-p)^2
19+p+.25(256-32p+p^2)
19-p+64-8p+.25p^2
83-9p+.25p^2

If this is right where do I go next?
Yes:

$TC=83-9p+0.25 p^2$

but you don't want to do that, what you want to do is write:

$p=16-Q$

Then:

${\rm{Profit}} = Q\times p - TC= Q(16-Q)-(3 + Q + 0.25 Q^2)=
-1.25 Q^2+15Q-3$

Now you need to find the $Q$ that maximises ${\rm{Profit}}$ using whatever method you have been shown for maximising a quadratic.

CB