A sporting goods store uses quadratic equations to monitor the daily cost and profit for various items it sells. The store's daily profit, $y$, when soccer balls are sold at $x$ dollars each, is modeled by $y=-6 x^2+100 x-180$.
Why is there an interval over which the graph decreases?
A. If the store sells more soccer balls, they can decrease the price.
B. If the soccer balls are returned for a refund, the store will lose money.
C. If the soccer balls are too expensive, fewer will be sold, reducing profit.
Answer (2)
The profit function is a downward-opening parabola, indicating a maximum profit point.
Beyond the vertex (maximum profit), increasing the price leads to decreased sales and, consequently, lower profit.
The decreasing interval on the graph represents the scenario where the price is too high, resulting in fewer sales and reduced profit.
Therefore, the correct answer is: If the soccer balls are too expensive, fewer will be sold, reducing profit.
Explanation
Understanding the Profit Function The problem describes a quadratic function representing the daily profit y from selling soccer balls at a price x . The question asks why the graph of this profit function decreases over a certain interval. The key to understanding this lies in recognizing that the quadratic function has a parabolic shape that opens downwards due to the negative coefficient of the x 2 term. This means the profit increases to a maximum point (the vertex) and then decreases as the price increases further.
Analyzing the Quadratic Function The profit function is given by y = − 6 x 2 + 100 x − 180 . This is a quadratic function in the form y = a x 2 + b x + c , where a = − 6 , b = 100 , and c = − 180 . Since a < 0 , the parabola opens downwards, indicating that there is a maximum profit.
Relating Price and Profit The vertex of the parabola represents the price at which the profit is maximized. To the right of the vertex, as the price x increases, the profit y decreases. This is because if the soccer balls are priced too high, fewer people will buy them, leading to a reduction in profit.
Conclusion Therefore, the reason the graph decreases over an interval is that if the soccer balls are too expensive, fewer will be sold, reducing profit.
Examples
Consider a small bakery that sells cakes. If they price their cakes too low, they might sell a lot, but their profit per cake is small, so their overall profit might not be great. If they price their cakes too high, they might not sell many, and again, their overall profit is low. There's an optimal price point where they maximize their profit. This problem illustrates how businesses use mathematical models to understand the relationship between price and profit and make informed decisions.
The profit function decreases over an interval because, after a certain point, higher prices lead to fewer sales. This means that the overall profit declines despite the increased price. The correct choice is option C: "If the soccer balls are too expensive, fewer will be sold, reducing profit."
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