Question

3) A small shoe company models its profit, y, in thousands of dollars, as a function of the number of thousands

of pairs of shoes sold, x. The function is y=-1000x² + 12000x - 11000.
a) How many pairs of shoes must be sold for the company to make $0 in profit?

Answer 1

To break even and earn $0 in profit, the shoe company must sell 10,000 pairs of shoes. The number was determined by solving the quadratic profit function for when profit equals zero using the quadratic formula.

Step-by-step explanation:

To determine how many pairs of shoes need to be sold for the company to make $0 in profit, we need to set the profit function y = -1000x² + 12000x - 11000 equal to zero and solve for x.

This involves finding the roots of the quadratic equation, which can be done by factoring, completing the square, or using the quadratic formula.

In this case, we will use the quadratic formula which is x = [-b ± √(b² - 4ac)]/(2a), where a = -1000, b = 12000, and c = -11000.

The quadratic formula yields:
x = [12000 ± √(12000² - 4 * -1000 * -11000)] / (2 * -1000)
x = [12000 ± √(144000000 - 44000000)] / -2000
x = [12000 ± √(100000000)] / -2000
x = [12000 ± 10000] / -2000
This gives us two possible values for x, but since we cannot sell a negative number of shoes, we disregard the negative root.
Thus, x = [12000 - 10000] / -2000
x = 2000 / -2000
x = -1

The positive number of shoes that must be sold for the company to break even (i.e., make $0 in profit) are 10,000 pairs.