the price that a company charged for a basketball hoop is given by the equation 50 - 5x² where x is the…

Explanation:

Step1: Define the profit formula

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

Step2: Find the value of $x$ when current production is 1 million

When $x = 1$ million, $P=(20\times1-5\times1^{3})=15$ million dollars.

Step3: Set up the equation for the previous production

We want to find $x$ such that $20x-5x^{3}=15$. Rearrange the equation to get $5x^{3}-20x + 15 = 0$. Divide through by 5: $x^{3}-4x + 3=0$.

Step4: 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). For the quadratic factor $x^{2}+x - 3=0$, use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ with $a = 1$, $b = 1$, and $c=-3$. $x=\frac{-1\pm\sqrt{1^{2}-4\times1\times(-3)}}{2\times1}=\frac{-1\pm\sqrt{1 + 12}}{2}=\frac{-1\pm\sqrt{13}}{2}$. We take the positive root since $x$ (number of hoops) cannot be negative. $x=\frac{-1+\sqrt{13}}{2}\approx\frac{-1 + 3.606}{2}\approx1.3$ million hoops.

Answer:

1.3 million hoops