Question

A computer program is gradually increasing the area of a square on the screen. Every second, the area of the square increases by 1 〖cm〗^2. The x-values in the following table represent the time in seconds. The y-values represent the approximate side length of the square shown on the screen at that time.
x (time in seconds) y (side length of square in centimeters)
0 0
1 1
2 1.4142
3 1.7321
4 2
5 2.2361

Find the average rate of change between x=1 and x=3. Interpret your result.

Answer 1

To find the average rate of change of the side length of the square (y-values) between \(x = 1\) and \(x = 3\), we can use the formula for the average rate of change:

\[
\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}
\]

In this case, we have:
- \(x_1 = 1\), \(y_1 = 1\) (side length at 1 second)
- \(x_2 = 3\), \(y_2 \approx 1.7321\) (side length at 3 seconds)

Now substituting these values into the formula:

\[
\text{Average Rate of Change} = \frac{1.7321 - 1}{3 - 1} = \frac{0.7321}{2} \approx 0.36605
\]

Thus, the average rate of change of the side length of the square between \(x = 1\) and \(x = 3\) seconds is approximately \(0.36605 \, \text{cm/second}\).

### Interpretation:
This means that, on average, the side length of the square is increasing by approximately \(0.36605\) centimeters per second during the interval from 1 second to 3 seconds. This average rate indicates how quickly the square's side length is growing as the area increases by \(1 \, \text{cm}^2\) each second. Since the area is increasing uniformly, the side length growth accelerates as the area increases, aligning with properties of squares where an increase in area translates to a nonlinear increase in side lengths (due to the square root relationship).