Radon Research Corporation (RRC)is one of 24 firms in Albuquerque testing homes for dangerous levels of radon gas.There is a standard test that all testing companies use.The manager of RRC wants to know the number of homes to test in 2021in order to maximize the firm's profit.The manager forecasted a price of $160 for radon tests in 2021.The firm's marginal cost was estimated as $S M C = 200 - 15 Q + 0.8 Q ^ { 2 }$ where Q is the number of tests performed each week.RRC's fixed cost will be $250 per week.The weekly profit (loss)at RRC in 2021will be

Question

Quizplus · Accepted Answer

To maximize profit, we need to find the quantity of tests that will give us the highest profit. For this, we need to find the quantity where marginal revenue (MR) equals marginal cost (MC).

We know that the price of a radon test in 2021 is $160. Assuming that the number of tests performed each week is Q, we can calculate the revenue as P × Q = $160Q.

To find the marginal revenue, we need to find the derivative of the revenue function with respect to Q. MR = d/dQ ($160Q) = $160.

Now, to find the profit function, we need to subtract the total cost from the revenue. The total cost includes both fixed cost and variable cost. The variable cost is given by the marginal cost function, which is $S M C = 200 - 15 Q + 0.8 Q ^ { 2 }$. So, the total cost function can be written as TC = $250 + $200Q - $15Q^2 + $0.4Q^3.

The profit function is given by π = TR - TC = ($160Q) - ($250 + $200Q - $15Q^2 + $0.4Q^3).

To find the quantity that maximizes profit, we need to take the derivative of the profit function with respect to Q and set it equal to zero:

dπ/dQ = $160 - $200 + $30Q - $1.2Q^2 = 0

Solving this equation, we get Q = 20 tests per week.

To check if this gives us a maximum or a minimum, we need to take the second derivative of the profit function with respect to Q:

d^2π/dQ^2 = $30 - $2.4Q

When Q = 20, d^2π/dQ^2 = $6, which is positive. This means that Q = 20 gives us a local minimum. Since the profit function is a downwards opening parabola, this local minimum is also the maximum.

Now, we plug Q = 20 back into the profit function to find the maximum profit:

π = ($160 × 20) - ($250 + $200 × 20 - $15 × 20^2 + $0.4 × 20^3) = $1,239.

Therefore, the weekly profit at RRC in 2021 will be $1,239.

Option D is the correct answer.

Radon Research Corporation (RRC)is One of 24 Firms in Albuquerque SMC=200−15Q+0.8Q2S M C = 200 - 15 Q + 0.8 Q ^ { 2 }SMC=200−15Q+0.8Q2

Radon Research Corporation (RRC)is One of 24 Firms in Albuquerque $S M C = 200 - 15 Q + 0.8 Q ^ { 2 }$