Question

2 the scatter plot shows the number of minutes people had to wait for service at a restaurant and the number of staff available at the time. a line that models the data is given by the equation ( y = - 1.62x + 18 ), where ( y ) represents the minutes of wait time, and ( x ) represents the number of staff available. a. the slope of the line is - 1.62. what does this mean in this situation? is it realistic? b. the ( y ) - intercept is ( (0,18) ). what does this mean in this situation? is it realistic?

Explanation:

Part a

• Meaning of slope: In the context of the linear equation \(y = mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept), the slope \(m=-1.62\). Here, \(y\) is the wait time (in minutes) and \(x\) is the number of staff available. The slope represents the rate of change. A negative slope means that as the number of staff (\(x\)) increases, the wait time (\(y\)) decreases. The value of \(- 1.62\) means that for each additional staff member available, the wait time decreases by approximately \(1.62\) minutes.
• Realism: This is realistic. More staff can handle more customers or tasks simultaneously, which should lead to a reduction in wait time. In a service - based business like a restaurant, adding more workers (within reasonable limits) can improve efficiency and decrease customer wait times.

Part b

• Meaning of y - intercept: The y - intercept occurs when \(x = 0\). In the equation \(y=-1.62x + 18\), when \(x = 0\) (i.e., there are no staff available), \(y=18\). So, it means that if there are no staff members available at the restaurant, the wait time is \(18\) minutes.
• Realism: This is not very realistic. If there are no staff members, there is no one to serve the customers, and it is more likely that the wait time would be indefinite (or the service would not be provided at all) rather than a fixed \(18\) - minute wait. In a real - world scenario, a restaurant with no staff cannot operate in a way that gives a predictable \(18\) - minute wait.

Answer:

Part a

• Meaning: For each additional staff member, the wait time decreases by approximately \(1.62\) minutes.
• Realism: Yes.

Part b

• Meaning: If there are no staff members, the wait time is \(18\) minutes.
• Realism: No.