A coffee shop determines that the daily profit on scones obtained by charging \( s \) dollars per scone is \( P(s)=-20 s^{2}+160 s-30 \). The coffee shop currently charges \( \$ 3.25 \) per scone. Find \( P^{\prime}(3.25) \), the rate of change of profit when the price is \( \$ 3.25 \). \( P^{\prime}(3.25)=\$ \) Number per scone. Based on the value of \( P^{\prime}(3.25) \) the coffee shop should Click for List \( \quad \) its price.

To find \( P^{\prime}(3.25) \), we must first calculate the derivative of the profit function \( P(s) \). The profit function is given as: \[ P(s) = -20s^2 + 160s - 30 \] By differentiating with respect to \( s \): \[ P'(s) = \frac{d}{ds}(-20s^2 + 160s - 30) = -40s + 160 \] Now, substituting \( s = 3.25 \) into the derivative: \[ P'(3.25) = -40(3.25) + 160 = -130 + 160 = 30 \] Thus, \( P'(3.25) = 30 \) dollars per scone. This positive value indicates that the profit is increasing at this price. Therefore, based on \( P^{\prime}(3.25) \), the coffee shop should **consider increasing** its price. When it comes to pricing, it's important to know how demand responds. If a business raises prices, they might attract less volume, but sometimes the higher price compensates for that — this is known as the **price elasticity of demand**. So, while raising prices may seem counterintuitive, the right market conditions can make it profitable! Speaking of market conditions, did you know that many successful businesses regularly analyze competitor pricing and customer feedback? They rely on data to determine the optimal price point, balancing between maximizing profit and ensuring customer satisfaction. It's like a dance, staying in tune with both your costs and your customers' desires!