Economics- How do i calculate price, output and profit when profit is maximised?

Given this formula,
TC = 0.5Q^2 + 10Q
Q = 100 − P

(^2 = squared)


How can i calculate the firms output, price and profit when profit is maximised?

Really stuck!

Answer:
The total cost function implies the marginal cost functioon MC=dTC/dQ=Q+10.

The demand function implies the inverse demand function P=100-Q. Since this function is linear, the marginal revenue function has the same vertical intercept but the marginal function is twice as steep. That is, MR=100-2Q.

Setting MR=MC yields.

100-2Q=Q+10 or 3Q=90 or Q=30 units.

The monopolist can sell each of these units for the price P=100-30=$70/unit, so his revenue is P*Q=70(30)=$2,100.

The total cost of producing 30 units is TC=.5(30^2)+10(30)
=$750, so maximum profit is 2,100-750=$1,350.

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