To determine how many basketball hoops the company previously produced to make the same profit of [tex]$15$[/tex] million dollars, we can follow these steps:
1. Define Variables and Constants:
- Let [tex]\( x \)[/tex] be the number of basketball hoops produced in millions.
- The price per hoop, [tex]\( P(x) \)[/tex], is given by the equation: [tex]\( P(x) = 50 - 5x^2 \)[/tex] dollars.
- The cost per hoop is [tex]$30.
- The profit they maintained is $[/tex]15$ million.
2. Profit Formula:
- Profit is calculated as the difference between the total revenue and the total cost.
- Revenue is given by [tex]\( \text{Revenue} = (\text{Price per hoop}) \times (\text{Number of hoops produced}) \)[/tex].
- Total cost is given by [tex]\( \text{Cost} = (\text{Cost per hoop}) \times (\text{Number of hoops produced}) \)[/tex].
3. Set Up the Profit Equation:
- Using the profit formula, [tex]\( \text{Profit} = \text{Revenue} - \text{Cost} \)[/tex].
[tex]\[
15,000,000 = \left(50 - 5x^2\right)x - 30x \times 1,000,000
\][/tex]
4. Simplify the Profit Equation:
- Substitute the given conditions and solve for [tex]\( x \)[/tex]:
[tex]\[
\begin{aligned}
15,000,000 &= (50 - 5x^2)x \times 1,000,000 - 30x \times 1,000,000 \\
15 &= (50 - 5x^2)x - 30x \\
15 &= 50x - 5x^3 - 30x \\
15 &= 20x - 5x^3
\end{aligned}
\][/tex]
5. Solve the Cubic Equation:
- Rearrange the equation to get all terms on one side:
[tex]\[
5x^3 - 20x + 15 = 0
\][/tex]
6. Find the Roots:
Since we are solving for [tex]\( x \)[/tex] where [tex]\( x \)[/tex] represents millions of basketball hoops, we are looking for realistic, positive solutions to this equation.
7. Approximate Solution:
We will approximate [tex]\( x \)[/tex] which makes the equation true:
By inspection or using numerical techniques, we can identify that the realistic solution is close to [tex]\( 1.3 \)[/tex] or [tex]\( 1.4 \)[/tex].
After solving the cubic equation, we find that the reasonable solution among the given options is:
[tex]\[ x \approx 1.4 \][/tex]
Therefore, the company previously produced approximately 1.4 million hoops.
