- The profit function \(y = -6x^2 + 100x - 180\) maps price \(x\) to profit \(y\).
- Statement 1 is false because \(y=170\) is the profit, not the quantity of items sold.
- Statement 2 is true because evaluating the profit function at \(x=8\) yields \(\$236\).
- Statement 3 is false because the profit function is a downward-opening parabola that decreases after reaching its vertex at \(x \approx 8.33\).
- The true statement is: \(\boxed{\text{If the store charges \$8 per soccer ball, it makes a daily profit of \$236.}}\)
Explanation
- Problem Analysis
The profit model is given by the quadratic equation \(y = -6x^2 + 100x - 180\), where \(x\) is the selling price in dollars and \(y\) is the daily profit in dollars. We are provided with a set of data points that satisfy this equation: \(x=5 \implies y=170\), \(x=8 \implies y=236\), and \(x=12 \implies y=156\). We need to evaluate three statements to identify the true one.
- Evaluating Statement 1
Statement 1 claims that if the price is \(\$5\), the store sells \(170\) balls. However, the model defines \(y\) as the profit in dollars, not the quantity of items sold. For \(x=5\), the calculated profit is \(y = -6(5)^2 + 100(5) - 180 = -150 + 500 - 180 = 170\). This confirms the profit is \(\$170\), not the number of units sold. Therefore, this statement is false.
- Evaluating Statement 2
Statement 2 claims that if the store charges \(\$8\) per ball, it makes a profit of \(\$236\). Substituting \(x=8\) into our profit equation:
\[y = -6(8)^2 + 100(8) - 180 = -6(64) + 800 - 180 = -384 + 800 - 180 = 236.\]
This matches the provided table and confirms the statement is true.
- Evaluating Statement 3
Statement 3 claims that higher selling prices always lead to higher profit. Since the equation is a parabola opening downward (the coefficient of \(x^2\) is negative), the profit increases until it reaches the vertex and then decreases. The vertex occurs at \(x = \frac{-b}{2a} = \frac{-100}{2(-6)} = \frac{100}{12} \approx 8.33\). Since the profit decreases for prices higher than approximately \(\$8.33\), the claim that profit always increases with price is false.
- Conclusion
Based on the evaluations, the only true statement is that if the store charges \(\$8\) per ball, it makes a daily profit of \(\$236\). The final answer is \(\boxed{\text{If the store charges \$8 per soccer ball, it makes a daily profit of \$236.}}\).
Examples
This type of analysis is crucial for business owners and retail managers to optimize their pricing strategies. By modeling profit as a function of price, they can identify the "sweet spot"—the price point that maximizes their daily earnings. Just like in the soccer ball example, understanding that profit doesn't just infinitely increase with price allows managers to avoid overpricing, which can decrease demand and lead to lower overall profit. Mathematical models provide the necessary evidence to make informed decisions that balance sales volume and profit margins.