Thank you for visiting A coffee shop determines that the daily profit on scones obtained by charging tex s tex dollars per scone is tex P s 20s 2. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
Final answer:
The rate of change of profit when the price per scone is $3.25 is positive. Therefore, increasing the price of scones could lead to an increase in profit, but other factors such as customer demand should be considered.
Explanation:
The first step is to find the derivative of P(s), which we can denote as P'(s). The derivative of a function represents the rate of change of that function with respect to its variable. So in this case, P'(s) will reveal how the coffee shop's profit changes as the price per scone changes. Here's how to calculate it:
Given the function P(s) = -20s² + 150s - 10, the derivative P'(s) = -40s + 150.
Next, we need to substitute s = 3.25 into P'(s) to find the rate of change of the profit when the price is $3.25. P'(3.25) = -40 * 3.25 + 150 = -130 + 150 = 20.
Because P'(3.25) is positive, this indicates that when the price is $3.25, an increase in price would also increase the profit, so the coffee shop might want to consider raising its prices on scones. However, it's important to consider other factors as well, such as customer demand, as a significant price increase might decrease the quantity of scones sold.
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