- The profit function is defined as \(y = -6x^2 + 100x - 180\).
- To find the zeros, set \(y = 0\) and use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with coefficients \(a = -6, b = 100, c = -180\).
- The discriminant is \(b^2 - 4ac = 100^2 - 4(-6)(-180) = 5680\).
- The calculated zeros are approximately \(x \approx 2.05\) and \(x \approx 14.61\), which correspond to options \(\boxed{c) \text{ and } d)}\).
Explanation
- Problem Analysis
To find the zeroes of the quadratic function \(y = -6x^2 + 100x - 180\), we need to solve for the values of \(x\) that make the profit \(y\) equal to zero. This leads to the quadratic equation \(-6x^2 + 100x - 180 = 0\). We identify the coefficients as \(a = -6\), \(b = 100\), and \(c = -180\).
- Applying the Quadratic Formula
We use the quadratic formula, which is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
First, we calculate the discriminant \(D = b^2 - 4ac\):
\[D = (100)^2 - 4(-6)(-180) = 10000 - 4320 = 5680\]
Next, we find the square root of the discriminant:
\[\sqrt{5680} \approx 75.3658\]
Now we substitute these values into the quadratic formula:
\[x = \frac{-100 \pm 75.3658}{2(-6)} = \frac{-100 \pm 75.3658}{-12}\]
- Final Calculation and Identification of Zeros
Now we calculate the two possible values for \(x\):
For the first root:
\[x_1 = \frac{-100 + 75.3658}{-12} = \frac{-24.6342}{-12} \approx 2.05285 \approx 2.05\]
For the second root:
\[x_2 = \frac{-100 - 75.3658}{-12} = \frac{-175.3658}{-12} \approx 14.6138 \approx 14.61\]
Comparing these results to the given options, we find that the values \(2.05\) and \(14.61\) correspond to options c) and d).
Examples
Understanding how to find the zeroes of a profit function is essential in economics and business management. It helps determine the 'break-even points'—the selling prices at which the company neither makes a profit nor incurs a loss. For instance, if a company sells soccer balls, calculating these prices allows managers to identify the range of pricing that keeps the business profitable. Knowing the range between these two zero-profit points, in this case, between approximately \(\$2.05\) and \(\$14.61\), guides strategic pricing decisions to maximize gains.