Question

the price that a company charged for a basketball hoop is given by the equation \\(50 - 5x^2\\) where \\(x\\) is the number of hoops that are produced, in millions. it costs the company \\(\\$30\\) to make each basketball hoop. the company recently reduced its production to 1 million hoops but maintained its profit of 15 million dollars. approximately how many basketball hoops did the company previously produce to make the same profit?

1.3 million hoops
1.4 million hoops
15 million hoops
30 million hoops

Explanation:

Formulate the profit function

Using the Revenue and Cost Functions and Profit Optimization knowledge points

$$
LATEXBLOCK0
$$

Set up the equation for the given profit

Using the Polynomial Equation Solving knowledge point

$$
LATEXBLOCK1
$$

Solve the cubic equation

Using the Polynomial Equation Solving knowledge point

$$
LATEXBLOCK2
$$

Identify the previous production level

The problem states that the company recently reduced its production to \(1\) million hoops (\(x = 1\)) while maintaining a profit of \(15\) million dollars. The other positive production level that yields the same profit is:

$$ x \approx 1.3 \text{ million hoops} $$

Answer:

• (A) 1.3 million hoops (Correct answer)
• (B) 1.4 million hoops
• (C) 15 million hoops
• (D) 30 million hoops