The graph illustrates the relationship between the area and radius of a circle, with the function r(A) plotted. The curve shows how the radius changes as the area increases.

To find the approximate average rate of change in the radius as the area increases, we need to look at how much the radius changes compared to how much the area changes over the given interval.

Step 1: Identify the starting and ending points from the graph.
The problem asks for the rate of change as the area increases from 1 square foot to 5 square feet. We need to read the corresponding radius values ($y$-axis) for these area values ($x$-axis).

* At Area = 1: Look at $x = 1$ on the horizontal axis. Go up to the curve. The point on the graph aligns with $y = 0.5$ on the vertical axis. So, when the area is 1, the radius is approximately 0.5 feet.
* At Area = 5: Look at $x = 5$ on the horizontal axis. Go up to the curve. The point on the graph aligns with $y = 1.25$ on the vertical axis (it is halfway between 1.2 and 1.3). So, when the area is 5, the radius is approximately 1.25 feet.

Step 2: Use the average rate of change formula.
The average rate of change is calculated like a slope:
$$ \text{Average Rate of Change} = \frac{\text{Change in Radius}}{\text{Change in Area}} $$
$$ \text{Average Rate of Change} = \frac{r_2 - r_1}{A_2 - A_1} $$

Where:
* $r_2 = 1.25$ (final radius)
* $r_1 = 0.5$ (initial radius)
* $A_2 = 5$ (final area)
* $A_1 = 1$ (initial area)

Step 3: Perform the calculation.
$$ \text{Change in Radius} = 1.25 - 0.5 = 0.75 \text{ feet} $$
$$ \text{Change in Area} = 5 - 1 = 4 \text{ square feet} $$

Now, divide the change in radius by the change in area:
$$ \text{Rate} = \frac{0.75}{4} $$

$$ 0.75 \div 4 = 0.1875 $$

Step 4: Compare with the options.
The calculated value is $0.1875$. Let's look at the given choices:
1) $0.17$
2) $0.21$
3) $0.3$
4) $8$

The value $0.1875$ is closest to $0.17$ among the smaller decimal options? Let's re-read the graph carefully.
Maybe the point at $x=5$ is slightly lower?
If $r(5)$ was $1.2$, then $\frac{1.2 - 0.5}{4} = \frac{0.7}{4} = 0.175$. This is very close to $0.17$.
If $r(5)$ was $1.3$, then $\frac{1.3 - 0.5}{4} = \frac{0.8}{4} = 0.2$. This is close to $0.21$.

Let's look really closely at the grid.
At $x=1$, $y$ is exactly on the line for $0.5$.
At $x=5$, the curve passes through the intersection of grid lines? No.
The vertical grid lines are every 1 unit. The horizontal grid lines are every $0.1$ units (since there are 10 subdivisions between 0 and 1, wait, no. Between 0 and 1 there are 5 subdivisions? No, let's count. 0, 0.2, 0.4, 0.6... No.
Let's count the major grid lines on the y-axis: 0, 0.5, 1, 1.5, 2.
Between 0 and 0.5, there are 2 intermediate lines. That means each small grid square height is $0.5 / ? $. Let's look at the labels. 0, 0.5, 1, 1.5, 2, 2.5.
There are 5 intervals between 0 and 0.5? No, looking at the image, between 0 and 0.5 there are 2 grid lines. That would make them $0.166$? Unlikely.
Let's assume standard grid. Usually, it's 0.1 or 0.2.
Looking at the y-axis labels: 0, 0.5, 1, 1.5, 2, 2.5.
Between 0 and 0.5, there appear to be 4 small squares? Or 5?
Let's assume there are 5 subdivisions between 0 and 0.5. Then each line is $0.1$.
If each line is $0.1$:
At $x=1$, $y$ is on the 5th line up? No, it looks like it's on the line labeled 0.5. Yes.
At $x=5$, $y$ is above the line labeled 1. It is two small lines above 1. If each small line is 0.1, then $y = 1.2$.
Let's recalculate with $y=1.2$.
$\text{Rate} = \frac{1.2 - 0.5}{5 - 1} = \frac{0.7}{4} = 0.175$.

Rounding $0.175$ to two decimal places gives $0.18$.
Option 1 is $0.17$.
Option 2 is $0.21$.

$0.175$ is closer to $0.17$ than $0.21$?
$|0.175 - 0.17| = 0.005$
$|0.175 - 0.21| = 0.035$
So $0.17$ is the much better approximation.

Let's double check the reading at $x=5$.
Is it possible the radius is slightly higher?
The function is $r = \sqrt{\frac{A}{\pi}}$.
$r(1) = \sqrt{\frac{1}{\pi}} \approx 0.564$.
$r(5) = \sqrt{\frac{5}{\pi}} \approx 1.261$.
Actual Average Rate of Change = $\frac{1.261 - 0.564}{4} = \frac{0.697}{4} \approx 0.174$.

The theoretical mathematical answer is approximately $0.174$.
This rounds to 0.17.

Therefore, the first option is the correct one.

Final Answer: 0.17 feet per square foot