Max is painting a fence. The table shows the number of square feet he painted after various numbers of minutes. What is the rate of change?
| Minutes | Square Feet Painted |
| ------- | ------------------- |
| 1 | 3.5 |
| 2 | 7 |
| 3 | 10.5 |
| 4 | 14 |
| 5 | 17.5 |
A. -3.5
B. -1
C. 1
D. 3.5
Answer (2)
Calculate the rate of change between two points: 2 − 1 7 − 3.5 = 3.5 .
Verify the rate of change with another set of points: 3 − 2 10.5 − 7 = 3.5 .
The rate of change is constant at 3.5 square feet per minute.
The rate of change is 3.5 .
Explanation
Understanding the Problem We are given a table that shows the number of square feet Max painted after various numbers of minutes. We need to find the rate of change, which represents how many square feet Max paints per minute.
Calculating the Rate of Change To find the rate of change, we can calculate the change in square feet divided by the change in minutes between any two points in the table. Let's calculate the rate of change between the first two points (1 minute, 3.5 square feet) and (2 minutes, 7 square feet).
Determining the Rate of Change The rate of change is calculated as follows: Change in minutes Change in square feet = 2 − 1 7 − 3.5 = 1 3.5 = 3.5 So, the rate of change is 3.5 square feet per minute.
Verifying the Rate of Change We can verify this by calculating the rate of change between other points. For example, between (2 minutes, 7 square feet) and (3 minutes, 10.5 square feet): 3 − 2 10.5 − 7 = 1 3.5 = 3.5 This confirms that the rate of change is constant.
Final Answer Therefore, the rate of change is 3.5 square feet per minute.
Examples
Understanding rate of change is crucial in many real-life situations. For example, if you are tracking the distance you travel over time, the rate of change is your speed. If Max continues to paint at this rate, we can predict how much fence he will paint in a given amount of time. This concept is also used in business to analyze sales trends or in science to measure reaction rates.
Max paints the fence at a consistent rate of 3.5 square feet per minute. This was determined by calculating the change in square feet painted over the change in minutes using points from the provided table. The correct answer option is D. 3.5.
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