What can you conclude from her work? Check all that apply.
The function is continuous.
Time represents the dependent variable.
The scenario is represented by a linear function, since the rate of change is constant.
As the amount of time continues, there are fewer cups of juice poured per hour.
For every additional hour, 53 cups of juice are poured.
Answer (2)
The conclusions drawn from the student's work indicate that the function is continuous, it represents a linear function due to a constant rate of change, and for every additional hour, 53 cups of juice are poured. However, it is incorrect to say that time is the dependent variable or that fewer cups are poured as time goes on.
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Calculate the rate of change: 5 − 4 181 − 128 = 53 cups per hour.
The function is continuous.
The scenario is represented by a linear function, since the rate of change is constant.
For every additional hour, 53 cups of juice are poured.
The correct statements are:
The function is continuous.
The scenario is represented by a linear function, since the rate of change is constant.
For every additional hour, 53 cups of juice are poured.
Explanation
Understanding the Problem We are given a table that shows the amount of juice (in cups) at different times (in hours). We need to analyze the given statements and determine which ones are correct.
Calculating the Rate of Change First, let's calculate the rate of change of the juice being poured. The rate of change is the change in the amount of juice divided by the change in time. From the table, we have two points: (4 hours, 128 cups) and (5 hours, 181 cups). So, the rate of change is: 5 − 4 181 − 128 = 1 53 = 53 cups per hour.
Analyzing the Statements Now, let's analyze each statement:
The function is continuous: Since we are dealing with a real-world scenario, we can assume that the amount of juice and time can take on any value within the given range. Therefore, the function is continuous.
Time represents the dependent variable: Time is the independent variable, and the amount of juice depends on the time. So, this statement is incorrect.
The scenario is represented by a linear function, since the rate of change is constant: We only have two data points, so we can only determine the rate of change between these two points. We cannot definitively say that the rate of change is constant throughout the entire scenario. However, based on the two points, the rate of change is constant, so it could be a linear function. We will assume it is linear for the purpose of this problem.
As the amount of time continues, there are fewer cups of juice poured per hour: The rate of change is positive (53 cups per hour), which means that as time increases, the amount of juice also increases. So, this statement is incorrect.
For every additional hour, 53 cups of juice are poured: This statement is correct because we calculated the rate of change to be 53 cups per hour.
Conclusion Based on our analysis, the correct statements are:
The function is continuous.
The scenario is represented by a linear function, since the rate of change is constant.
For every additional hour, 53 cups of juice are poured.
Examples
Understanding rates of change is crucial in many real-world scenarios. For example, if you're tracking the growth of a plant, the rate of change tells you how many inches the plant grows per day. Similarly, in economics, the rate of change can represent the inflation rate, indicating how much prices increase per year. In physics, it could be the speed of an object, showing how many meters it travels per second. Recognizing and calculating rates of change helps us make informed decisions and predictions in various fields.
