Question 5 (6 points) 3. Let \( P(x)=-0.3 x^{2}+60 x-500 \) be a profit function for a cable television company. It estimates \( x \) as the number of subscribers in thousands and the profit is measured in thousands of dollars. (Use algebra to solve all the problems. To receive credit, all work must be shown) \[ \begin{array}{l}\text { a. A break-even point is where profit is zero. Use algebra to find the break-even points. Give } \\ \text { an interpretation of what this means for the cable company. (6 points) }\end{array} \]

To find the break-even points for the profit function \( P(x) = -0.3x^2 + 60x - 500 \), we need to set the profit function equal to zero and solve for \( x \): \[ P(x) = 0 \] This gives us the equation: \[ -0.3x^2 + 60x - 500 = 0 \] To solve this quadratic equation, we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = -0.3 \), \( b = 60 \), and \( c = -500 \). First, we calculate the discriminant \( b^2 - 4ac \): \[ b^2 = 60^2 = 3600 \] \[ 4ac = 4 \cdot (-0.3) \cdot (-500) = 4 \cdot 0.3 \cdot 500 = 600 \] \[ b^2 - 4ac = 3600 - 600 = 3000 \] Now we can substitute \( a \), \( b \), and the discriminant into the quadratic formula: \[ x = \frac{-60 \pm \sqrt{3000}}{2 \cdot -0.3} \] Calculating \( \sqrt{3000} \): \[ \sqrt{3000} = \sqrt{100 \cdot 30} = 10\sqrt{30} \] Now substituting this back into the formula: \[ x = \frac{-60 \pm 10\sqrt{30}}{-0.6} \] This simplifies to: \[ x = \frac{60 \mp 10\sqrt{30}}{0.6} \] Dividing both terms in the numerator by \( 0.6 \): \[ x = 100 \mp \frac{100\sqrt{30}}{6} \] Calculating \( \frac{100\sqrt{30}}{6} \): \[ \frac{100\sqrt{30}}{6} \approx 40.82 \quad (\text{using } \sqrt{30} \approx 5.477) \] Thus, we have two potential break-even points: \[ x_1 = 100 - 40.82 \approx 59.18 \] \[ x_2 = 100 + 40.82 \approx 140.82 \] So the break-even points are approximately \( x \approx 59.18 \) and \( x \approx 140.82 \). ### Interpretation The break-even points indicate the number of subscribers (in thousands) at which the cable television company neither makes a profit nor incurs a loss. Specifically: - At approximately 59,180 subscribers, the company starts to make a profit after this point. - At approximately 140,820 subscribers, the company will again reach a break-even point, after which it will start to incur losses if the number of subscribers exceeds this point. This information is crucial for the company as it helps them understand the subscriber thresholds they need to maintain to ensure profitability.