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A firm has the total cost function TC=120+45Q-Q^2+0.4Q^3 and faces a demand curve given P=240-20p what is the profit

User Max O
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Explanation:

To find the profit of the firm, we need to first determine the quantity Q that maximizes the profit, and then use that quantity to find the price and profit.

The profit function can be written as:

π(Q) = TR(Q) - TC(Q)

where TR(Q) is the total revenue function and TC(Q) is the total cost function. We can write TR(Q) as:

TR(Q) = P(Q) * Q

where P(Q) is the price function, which is given as:

P(Q) = 240 - 20Q

So, the profit function becomes:

π(Q) = (240 - 20Q) * Q - (120 + 45Q - Q^2 + 0.4Q^3)

Simplifying this expression, we get:

π(Q) = -0.4Q^3 + 24.6Q^2 - 195Q + 120

To maximize the profit, we take the derivative of the profit function with respect to Q and set it equal to zero:

π'(Q) = -1.2Q^2 + 49.2Q - 195 = 0

Solving for Q using the quadratic formula, we get:

Q = (49.2 ± sqrt(49.2^2 - 4*(-1.2)(-195))) / (2(-1.2))

Q = 21 or Q = 32.5

Since the coefficient of the Q^3 term in the profit function is negative, the profit function has a maximum at Q = 32.5. Therefore, the firm should produce and sell 32.5 units of output.

To find the price that the firm should charge, we substitute Q = 32.5 into the demand function:

P = 240 - 20Q

P = 240 - 20(32.5)

P = 160

Therefore, the firm should charge a price of $160 per unit.

To find the profit at the optimal level of output, we substitute Q = 32.5 and P = 160 into the profit function:

π(Q) = -0.4Q^3 + 24.6Q^2 - 195Q + 120

π(32.5) = -0.4(32.5)^3 + 24.6(32.5)^2 - 195(32.5) + 120

π(32.5) = $1,722.81

Therefore, the profit at the optimal level of output is $1,722.81

User Andrew Chelix
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