Question

independent practice
compare the increasing and decreasing linear functions described. then solve the problems. show your work.
1 a local cafe sells 7,200 cold beverages in the month of july. over the next six months, the number of cold beverages decreases at an average rate of 750 cups per month. during the same time frame, the number of hot beverages sold is represented by the equation ( y = 900x + 3,300 ) where the number of hot beverages, ( y ), is a function of ( x ), the number of months after july.
a. what is the rate of change of cold beverages sold per month? hot beverages?
b. which rate of change is greater? what does it mean?

Explanation:

Part a

Step1: Determine cold beverages' rate

The number of cold beverages sold in July is 7,200, and over 6 months, it decreases at 750 cups per month. The rate of change (slope) for a linear function is the change in quantity over change in time. For cold beverages, since it's decreasing, the rate is -750 cups per month (the negative sign indicates a decrease).

Step2: Determine hot beverages' rate

The equation for hot beverages is \( y = 900x + 3300 \). In the slope - intercept form of a linear equation \( y=mx + b \), \( m \) is the slope (rate of change). Here, \( m = 900 \), so the rate of change for hot beverages is 900 cups per month (the positive sign indicates an increase).

To find which rate of change is greater, we compare the absolute values (since rate of change can be negative for decrease and positive for increase, but we are looking at the magnitude of the change). The absolute value of the rate of change of cold beverages is \(| - 750|=750\), and the absolute value of the rate of change of hot beverages is \(|900| = 900\). Since \(900>750\), the rate of change of hot beverages is greater. In the context of the problem, a rate of change of 900 for hot beverages means that the number of hot beverages sold increases by 900 cups each month after July, while the number of cold beverages sold decreases by 750 cups each month after July.

Answer:

• Rate of change of cold beverages: \(-750\) cups per month (decreasing at 750 cups per month).
• Rate of change of hot beverages: \(900\) cups per month (increasing at 900 cups per month).

Part b