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.How many radon tests per week should be undertaken?

Question

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To maximize profit, a firm sets its marginal revenue (MR) equal to its marginal cost (MC). Given the price is constant at $160, the MR is also $160 for each additional unit sold because the additional revenue from selling one more test is constant. The marginal cost (MC) is given by the derivative of the total cost function, which is already provided as $SMC = 200 - 15Q + 0.8Q^2$. Setting MR equal to MC gives:$$160 = 200 - 15Q + 0.8Q^2$$Rearranging and solving for Q:$$0.8Q^2 - 15Q + 40 = 0$$Using the quadratic formula where $a = 0.8$, $b = -15$, and $c = 40$:$$Q = \frac{-(-15) \pm \sqrt{(-15)^2 - 4 \cdot 0.8 \cdot 40}}{2 \cdot 0.8}$$$$Q = \frac{15 \pm \sqrt{225 - 128}}{1.6}$$$$Q = \frac{15 \pm \sqrt{97}}{1.6}$$$$Q = \frac{15 \pm 9.8489}{1.6}$$Choosing the positive root for a positive quantity of tests gives:$$Q = \frac{15 + 9.8489}{1.6} \approx 15.5$$Therefore, RRC should undertake approximately 15.5 radon tests per week to maximize profit.

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 $S M C = 200 - 15 Q + 0.8 Q ^ { 2 }$