Question

The price that a company charged for a basketball hoop is given by the equation 50 minus 5 x squared 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?

Answer 1

The company should sell 1.3 million hoops to maintain the same profit

How to determine the number of hoops?

From the question x represents the number of hoops.

So, the functions are given as:

S(x)= 50 - 5x^2 ---- Price function

C(x)= 30x ---- cost function

The profit is calculated as:

P(x) = S(x) * Number of hoops - C(x)

This gives

P(x) = S(x) * x - C(x)

So, we have:

P(x) = (50 - 5x^2) * x- 30x

Expand

P(x) = 50x - 5x^3 - 30x

Evaluate the like terms

P(x) = 20x - 5x^3

When the profit is 15 million dollars, we have:

20x - 5x^3 = 15

Divide through by 5

4x - x^3 = 3

Rewrite as:

x^2 - 4x + 3 = 0

Using a graphing calculator, we have:

x = 1 or x = 1.3

Hence, the company should sell 1.3 million hoops to maintain the same profit

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