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. Simplify the profit formula: $P=(20 - 5x^{2})x=20x-5x^{3}$.

Step2: Substitute the known values

We know that when $x = 1$ (1 million hoops), $P = 15$ million. 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 follows: $x^{3}-4x + 3=(x - 1)(x^{2}+x - 3)$. We already know $x = 1$ is one solution. To find the other non - 1 solution, 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 the 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 take the positive root since the number of hoops cannot be negative. $x=\frac{-1+\sqrt{13}}{2}\approx\frac{-1 + 3.606}{2}=\frac{2.606}{2}=1.303\approx1.3$.

Answer:

1.3 million hoops