The problem states that 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 curve $SMC = q^2 + 10q + 10$. We are asked to determine the quantity that will be produced at a price of $49$.

1. Problem Description

The problem states that 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 curve $SMC = q^2 + 10q + 10$. We are asked to determine the quantity that will be produced at a price of $49$.

2. Solution Steps

In a perfectly competitive industry, a firm maximizes profit by producing at the level where price equals marginal cost. Therefore, we need to set the price equal to the short-run marginal cost and solve for $q$.

$Price = SMC$

$49 = q^2 + 10q + 10$

Subtract 49 from both sides:

$q^2 + 10q + 10 - 49 = 0$

$q^2 + 10q - 39 = 0$

We need to solve this quadratic equation for $q$. We can use the quadratic formula or try to factor the equation. Let's try to factor:

$(q + 13)(q - 3) = 0$

This gives us two possible solutions for $q$: $q = -13$ and $q = 3$.

Since quantity cannot be negative, we discard the solution $q = -13$.

Therefore, the quantity produced is $q = 3$.

3. Final Answer

The quantity produced is

3.