To determine the rate of change in the number of square feet Max painted per minute, we need to analyze the increments in time and how these increments affect the painted area. Let's break it down step-by-step:
1. Identify the given data:
The table shows the number of square feet painted after various numbers of minutes:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Minutes} & \text{Square Feet Painted} \\
\hline 1 & 3.5 \\
\hline 2 & 7 \\
\hline 3 & 10.5 \\
\hline 4 & 14 \\
\hline 5 & 17.5 \\
\hline
\end{array}
\][/tex]
2. Determine the changes in time and the corresponding changes in painted area:
- From minute 1 to minute 2:
- Time increment, [tex]\(\Delta x = 2 - 1 = 1\)[/tex] minute
- Painted increase, [tex]\(\Delta y = 7 - 3.5 = 3.5\)[/tex] square feet
- It's helpful to see if similar intervals hold for other pairs:
- From minute 2 to minute 3:
- Time increment, [tex]\(\Delta x = 3 - 2 = 1\)[/tex] minute
- Painted increase, [tex]\(\Delta y = 10.5 - 7 = 3.5\)[/tex] square feet
- From minute 3 to minute 4:
- Time increment, [tex]\(\Delta x = 4 - 3 = 1\)[/tex] minute
- Painted increase, [tex]\(\Delta y = 14 - 10.5 = 3.5\)[/tex] square feet
- From minute 4 to minute 5:
- Time increment, [tex]\(\Delta x = 5 - 4 = 1\)[/tex] minute
- Painted increase, [tex]\(\Delta y = 17.5 - 14 = 3.5\)[/tex] square feet
3. Calculate the rate of change:
The rate of change is defined as the change in painted area divided by the change in time. Using the increments calculated:
[tex]\[
\text{Rate of change} = \frac{\Delta y}{\Delta x} = \frac{3.5 \text{ square feet}}{1 \text{ minute}} = 3.5 \text{ square feet per minute}
\][/tex]
Therefore, the rate of change in the number of square feet Max painted per minute is [tex]\(3.5\)[/tex] square feet per minute.
