Final answer:
To earn a profit, the shoe company can sell any number of shoes except for exactly 4,000 pairs, as the profit function P(x) indicates that sales of less or more than this amount will still result in profit, but not the maximum profit.
Step-by-step explanation:
The profit of a shoe company is modeled by the function P(x) = -5(x – 4)2 + 45, where x is the number of pairs of shoes produced in thousands, and P(x) is the profit, in thousands of dollars. To determine how many thousands of pairs of shoes will the company need to sell to earn a profit, we are looking for the value(s) of x that make P(x) greater than zero.
To find when P(x) is greater than zero, set the inequality P(x) > 0, which gives us the inequality -5(x – 4)2 + 45 > 0. Solving for x, we find the critical point at x = 4 by setting the equation to zero -5(x – 4)2 + 45 = 0 and solving for x. Since the equation is a downward opening parabola (because of the -5 coefficient), the profit is positive on either side of the vertex.
The vertex, (4, 45), represents the maximum profit. Therefore, to make a profit, the company can sell any number of shoes except exactly 4,000 since this amount gives the maximum profit, and selling less or more would still result in profit, although less than the maximum.