the price that a company charged for a basketball hoop is given by the equation 50 - 5x^2 where x is the…

Explanation:

Step1: Define profit formula

Profit $P=(50 - 5x^{2}-30)x$. The cost per hoop is $30$, and the selling - price per hoop is $50 - 5x^{2}$, and $x$ is the number of hoops in millions. Simplify the profit formula: $P=(20 - 5x^{2})x=20x-5x^{3}$.

Step2: Substitute known values

We know that when $x = 1$ (1 million hoops), $P = 15$ (15 million dollars). We want to find $x$ when $P = 15$. So we set up the equation $15=20x-5x^{3}$, or $5x^{3}-20x + 15 = 0$. Divide through by 5 to get $x^{3}-4x + 3=0$.

Step3: Factor the equation

We can factor $x^{3}-4x + 3$ as $(x - 1)(x^{2}+x - 3)=0$. We already know $x = 1$ is a solution (the current production level). To find the other solutions, we use the quadratic formula for $x^{2}+x - 3=0$. The quadratic formula for $ax^{2}+bx + c = 0$ is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a = 1$, $b = 1$, and $c=-3$.

Step4: Apply quadratic formula

$x=\frac{-1\pm\sqrt{1^{2}-4\times1\times(-3)}}{2\times1}=\frac{-1\pm\sqrt{1 + 12}}{2}=\frac{-1\pm\sqrt{13}}{2}$. We get two solutions for the quadratic part: $x=\frac{-1+\sqrt{13}}{2}\approx\frac{-1 + 3.606}{2}\approx1.303$ and $x=\frac{-1-\sqrt{13}}{2}\approx\frac{-1 - 3.606}{2}\approx - 2.303$. Since the number of hoops $x$ (in millions) cannot be negative, we discard the negative solution.

Answer:

1.3 million hoops