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