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Calculate the average rate of change in the weight over the interval [4, 8].

To calculate the average rate of change of the man's weight \( W \) over the interval \([4, 8]\), we use the formula for the average rate of change of a function:

\[
\text{Average rate of change} = \frac{\Delta W}{\Delta x} = \frac{W(x_2) - W(x_1)}{x_2 - x_1}
\]

where \(x_1 = 4\), \(x_2 = 8\), \(W(x_1) = 36\), and \(W(x_2) = 54\) based on the table provided.

Substituting these values into the formula:

\[
\text{Average rate of change} = \frac{54 - 36}{8 - 4} = \frac{18}{4} = 4.5
\]

Therefore, the average rate of change in weight over the interval \([4, 8]\) is **4.5 pounds per year**.

Would you like to explore this further or have any questions?

Here are some related questions you might find interesting:
1. What is the average rate of change in weight over a different interval, such as \([8, 12]\)?
2. How can we interpret the average rate of change in a real-life context?
3. What is the significance of calculating the average rate of change in a function?
4. How does the average rate of change differ from the instantaneous rate of change?
5. Can we use the given data to predict the weight at other ages using linear approximation?

**Tip:** The average rate of change is a useful tool in understanding how a quantity varies over an interval. It provides a simple way to measure the overall change without considering the details of the change within the interval.

Learn how to calculate the average rate of change in weight over the interval [4, 8] using the formula ΔW / Δx = (W(8) - W(4)) / (8 - 4). This problem involves analyzing a linear function representing weight change over time. Suitable for students in Grades 9-12.

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