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 the profit - function

The revenue function $R(x)$ is the price per unit times the number of units. The price per unit is $p = 50 - 5x^{2}$, and the number of units is $x$ (in millions). So, $R(x)=x(50 - 5x^{2})=50x-5x^{3}$. The cost function $C(x)$ is the cost per unit times the number of units. Since the cost per unit is $30$, $C(x) = 30x$. The profit function $P(x)$ is $P(x)=R(x)-C(x)=(50x - 5x^{3})-30x=- 5x^{3}+20x$.

Step2: Set up the profit - equation

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

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 a solution (the current production level). We solve the quadratic equation $x^{2}+x - 3=0$ using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$, where $a = 1$, $b = 1$, 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 produced $x>0$. $x=\frac{-1+\sqrt{13}}{2}\approx\frac{-1 + 3.606}{2}=\frac{2.606}{2}=1.303\approx1.3$.

Answer:

1.3 million hoops