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?

Answer 1

Given:
Profit : 15,000,000
Cost: 30 per basketball hoop
production: 1 million hoops
price: 50 - 5x²

Profit = Sales - Cost
15,000,000 = sales - 30(1,000,000)
15,000,000 + 30,000,000 = sales
45,000,000 = sales

45,000,000 / 1,000,000 = 45 sales price.

Answer 2

The company produced 2 million basketball hoops in previous year.

Step-by-step explanation:

Let the number of basketball hoops produced by the company be x.

The equation of price is

`P=50-5x^2`

The cost of each basketball hoop is $30, so the cost function is

`C=30x`

The profit function is the difference between price and cost function.

`Profit=P(x)-C(x)`

`Profit=50-5x^2-30x`

The company reduced its production to 1 million hoops and make profit of 15 million.

`15=50-5(x-1)^2-30(x-1)`

`15=50-5(x^2-2x+1)-30x+30`

`15=50-5x^2+10x-5-30x+30`

`5x^2+20x-60=0`

`5(x^2+4x-12)=0`

`(x^2+6x-2x-12)=0`

`(x+6)(x-2)=0`

Therefore value of x is -6 and 2. The production can not be negative, therefore the company produced 2 million basketball hoops in previous year.