- Set up the break-even condition by equating revenue and expense functions: \(100p - 2p^2 = 700 - 10p\).
- Simplify the resulting quadratic equation: \(p^2 - 55p + 350 = 0\).
- Apply the quadratic formula to find the roots: \(p \approx 7.35\) and \(p \approx 47.66\).
- Match the calculated values to the nearest given option: \(\boxed{\$7 \text{ and } \$48}\).
Explanation
- Analysis of the Problem
The company's revenue function is given by \(R(p) = 100p - 2p^2\) and the expense function is given by \(E(p) = 700 - 10p\), where \(p\) represents the price per service. To find the break-even points, we need to determine the prices \(p\) where the revenue equals the expenses, which means solving the equation \(R(p) = E(p)\).
- Setting Up the Quadratic Equation
We set the two expressions equal to each other:
\[100p - 2p^2 = 700 - 10p\]
To solve for \(p\), we rearrange the equation into a standard quadratic form \(ap^2 + bp + c = 0\):
\[2p^2 - 110p + 700 = 0\]
We can simplify this equation by dividing all terms by 2:
\[p^2 - 55p + 350 = 0\]
- Solving for Price \(p\)
To solve the quadratic equation \(p^2 - 55p + 350 = 0\), we use the quadratic formula \(p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -55\), and \(c = 350\):
\[p = \frac{55 \pm \sqrt{(-55)^2 - 4(1)(350)}}{2(1)}\]
\[p = \frac{55 \pm \sqrt{3025 - 1400}}{2}\]
\[p = \frac{55 \pm \sqrt{1625}}{2}\]
Using the approximation \(\sqrt{1625} \approx 40.31\):
\[p_1 \approx \frac{55 - 40.31}{2} = \frac{14.69}{2} \approx 7.35\]
\[p_2 \approx \frac{55 + 40.31}{2} = \frac{95.31}{2} \approx 47.66\]
The break-even points are approximately \(p = 7.35\) and \(p = 47.66\). Comparing these results with the given options, they are closest to \(\$7\) and \(\$48\).
Examples
Understanding the break-even point is crucial for any business, such as a local café or a landscaping service. For example, if a bicycle shop knows its costs (like rent and parts) and its expected sales revenue for different bike models, they can calculate the minimum price they must charge to cover those costs. By setting revenue equal to expenses, business owners can determine their profit zones, ensuring they don't operate at a loss and can set competitive yet profitable prices. This calculation directly helps in financial planning and sustainable business management.