To find the measure of the central angle that cuts an arc of a given length on a circle, we can use the formula:
\[ \text{Arc Length} = r \times \theta \]
where \( \theta \) is the angle in radians and \( r \) is the radius of the circle.
First, we need to calculate the radius of the pool. The diameter of the circular fence around the pool is 20 feet, so the radius \( r \) is:
\[ r = \frac{\text{diameter}}{2} = \frac{20}{2} = 10 \text{ feet} \]
Next, we can set up the equation for arc length:
\[ 4 = 10 \times \theta \]
To solve for \( \theta \):
\[ \theta = \frac{4}{10} = 0.4 \text{ radians} \]
To convert radians to degrees, we use the conversion factor \( \frac{180}{\pi} \):
\[ \theta_{\text{degrees}} = 0.4 \times \frac{180}{\pi} \approx 22.9 \text{ degrees} \]
Thus, the measure of the central angle that cuts the arc that measures 4 feet is approximately 22.9 degrees.
So the answer is:
22.9 degrees