Answered

On Monday, a coffee shop owner gathers data about the number of coffees, $y$, sold during various hours, $x$, of the coffee shop being open. The table shows the data the owner gathers, which can be modeled by a quadratic function, where $0 \leq x \leq 15$.




Hour ( $x$ )		02561115

Number of Coffees ( $y$ )		09823625918587






If the trend continues the following Monday, approximately how many coffees should the coffee shop owner anticipate selling during the $8^{\text {th }}$ hour of being open?


A108 coffees
B150 coffees
C255 coffees
D272 coffees

Verified Answer :

  • Analyze the provided data points to identify the trend modeled by a quadratic function.
  • Determine the quadratic function coefficients \(a\), \(b\), and \(c\) using polynomial regression.
  • Calculate the number of coffees sold at \(x = 8\) using the derived function \(y \approx -3.85x^2 + 62.40x + 2.40\).
  • Conclude that the predicted number of coffees is approximately 255, matching option C.
  • Final Answer: \(\boxed{C}\)

Explanation

  1. Understanding the Problem and Analyzing the Data

We are given data for the number of coffees \(y\) sold in a coffee shop at different hours \(x\). The data points are (0, 0), (2, 98), (5, 236), (6, 259), (11, 185), and (15, 87). We are told that these data points can be modeled by a quadratic function, which takes the form \(y = ax^2 + bx + c\). Our objective is to determine the predicted number of coffees sold at the 8th hour (\(x = 8\)).

  1. Determining the Quadratic Model

To find the quadratic function that best models the given data, we determine the coefficients \(a\), \(b\), and \(c\) using a least-squares polynomial fit. This approach minimizes the sum of the squares of the vertical deviations between the data points and the quadratic function. By calculating this fit, we find the function that best approximates the provided points: (0, 0), (2, 98), (5, 236), (6, 259), (11, 185), and (15, 87). The resulting quadratic function is approximately \(y \approx -3.85x^2 + 62.40x + 2.40\).

  1. Calculating the Prediction and Conclusion

With the quadratic model \(y = -3.8518x^2 + 62.4019x + 2.4035\) determined, we now substitute \(x = 8\) into this expression to estimate the number of coffees sold. Performing the calculation:
\[y(8) = -3.8518(8)^2 + 62.4019(8) + 2.4035\]
\[y(8) = -3.8518(64) + 499.2152 + 2.4035\]
\[y(8) = -246.5152 + 499.2152 + 2.4035 \approx 255.10\]
This result indicates that the expected number of coffees sold at the 8th hour is approximately 255. Comparing this to the given options, we select option C.

Examples

In business, quadratic regression is often used to model trends that increase and then decrease, like daily sales cycles. By tracking sales at different times of the day, a store owner can predict peak hours and manage staffing or inventory levels efficiently. For example, if a shop owner knows that the 8th hour is a high-traffic period, they can ensure extra staff is scheduled to meet the increased demand.

Related Questions