We will use the demand function "P + 2Q = 50" to solve various problems .
From the function [tex]P + 2Q = 50[/tex]
We then rearrange to [tex]P = 50 - 2Q[/tex]
Let remember that Total Revenue = Price * Quantity
Total Revenue [tex]TR = P * Q[/tex]
[tex]TR = 50*Q - 2Q*Q[/tex]
[tex]TR = 50Q - 2Q2[/tex]
Let remember that Total cost = Fixed Cost + (Quantity*Variable cost)
[tex]TC = FC + [Q * (AVC)]\\TC = 30 + [Q * (Q + 3)]\\TC = 30 + Q2 + 3Q[/tex]
Now to derive profit, we will subtract Total cost from Total Revenue[tex]Profit = TR - TC \\Profit = 50Q - 2Q2 - 30 - Q2 - 3Q\\Profit = 47Q - 3Q2 - 30\\Profit = 3Q² + 47Q - 30[/tex]
In break-even, the profit equals 0. Z = 0
[tex]47Q - 3Q2 - 30 = 0\\3Q2 - 47Q + 30 = 0\\3Q2 - 45Q - 2Q + 30 = 0\\3Q(Q - 15) - 2(Q - 15) = 0\\(Q - 15) x (3Q - 2) = 0\\Q = 15 or Q = 2/3[/tex]
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