A chemical company in a perfectly competitive industry has a short-run total cost curve $TC = (1/3)q^3 + 5q^2 + 10q + 10$ and a short-run marginal cost $SMC = q^2 + 10q + 10$. We need to find the quantity produced at a price of 385.

1. Problem Description

A chemical company in a perfectly competitive industry has a short-run total cost curve $TC = (1/3)q^3 + 5q^2 + 10q + 10$ and a short-run marginal cost $SMC = q^2 + 10q + 10$. We need to find the quantity produced at a price of

3

8

5.

2. Solution Steps

In a perfectly competitive industry, a firm maximizes its profit by producing where the price equals the marginal cost. Therefore, we need to solve the equation $P = SMC$ for $q$.

Given that the price $P = 385$ and $SMC = q^2 + 10q + 10$, we have

$385 = q^2 + 10q + 10$.

Subtracting 385 from both sides, we get

$q^2 + 10q - 375 = 0$.

We can solve this quadratic equation for $q$. We are looking for two numbers that multiply to -375 and add to

1

0. These numbers are 25 and -

1

5. So, we can factor the quadratic as follows:

$(q + 25)(q - 15) = 0$.

Therefore, the solutions for $q$ are $q = -25$ and $q = 15$. Since quantity cannot be negative, we discard the negative solution. Therefore, the quantity produced is $q = 15$.

3. Final Answer

The quantity produced is

1

5. Final Answer: c. 15