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6. A firm uses a single plant with costs C = 160 + 16Q + .1Q2 and faces the price equation P = 96 - .4Q.a. Find the firm's profit-maximizing price

I have the answer for the question on page two, but I do not understand the math. I believe we take the derivative, but I am not sure. Can you explain how the answer was found, so I can understand it? Page 2 only, please. 

6. A frm uses a single plant with costs C = 160 + 16Q + .1Q 2 and Faces the price equaTon P = 96 °- .4Q.a. ±ind the frm’s proft-maximizing price and quanTty. What is its proft? MR = 96 - .8Q MC = 16 + .2Q 96 - .8Q = 16 + .2Q 80 = Q P = 96 - .4(80) P = 64 Proft = 80(64) – 160 – 16(80) - .1(80)^2 = 3040
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The frm’s producton manager claims ±ha± ±he frm’s average cos± oF producton is minimized a± an ou±pu± oF 40 uni±s. ²ur±hermore, she claims ±ha± 40 uni±s is ±he frm’s prof±-maximizing level oF ou±pu±. Explain whe±her ±hese claims are correc±. AC = 160/Q +16 +.1Q To minimise ±his dAC/dQ = 0 =-160/Q 2 +.1 Or Q = 40 Bu± a± ±his level prof±s are no± maximised. This happens a± Q= 80 only. So ±he claims are righ± abou± AC being maximised, bu± no± prof± maximisaton level oF Q
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6. A firm uses a single plant with costs C = 160 + 16Q + .1Q2 and faces the price
equation P = 96 - .4Q.a. Find the firm’s profit-maximizing price and quantity.
What is its profit?
MR = 96 - .8Q...

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