Suppose a chemical company is in a perfectly competitive industry and has a short run total cost curve of TC = 1/3q3 + 5q2 + 10q + 10 and a short run marginal cost of SMC = q2 + 10q + 10. At the price of 49, how many will be produced?Question 4Select one:a.3.b.15.c.0.d.5.
Question
Suppose a chemical company is in a perfectly competitive industry and has a short run total cost curve of TC = 1/3q3 + 5q2 + 10q + 10 and a short run marginal cost of SMC = q2 + 10q + 10. At the price of 49, how many will be produced?Question 4Select one:a.3.b.15.c.0.d.5.
Solution
In a perfectly competitive market, a firm will produce where the price equals the marginal cost (P = MC). Given the price (P) is 49, we set the short run marginal cost (SMC) equal to 49 and solve for q (quantity).
So, we have:
49 = q^2 + 10q + 10
Rearranging the equation, we get:
q^2 + 10q - 39 = 0
This is a quadratic equation in the form of ax^2 + bx + c = 0. We can solve for q using the quadratic formula, which is q = [-b ± sqrt(b^2 - 4ac)] / 2a.
Here, a = 1, b = 10, and c = -39.
Substituting these values into the quadratic formula, we get:
q = [-10 ± sqrt((10)^2 - 41(-39))] / 2*1 q = [-10 ± sqrt(100 + 156)] / 2 q = [-10 ± sqrt(256)] / 2 q = [-10 ± 16] / 2
This gives us two possible solutions: q = 3 or q = -13.
However, quantity cannot be negative in this context, so we discard -13. Therefore, the chemical company will produce 3 units when the price is 49. So, the correct answer is a. 3.
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