To find the radius of the lake, we can use the formula for the circumference of a circle, which is given by:
\[ C = 2\pi r \]
where \(C\) is the circumference and \(r\) is the radius. Given that the circumference \(C\) is 24 miles, we can solve for the radius \(r\):
\[ 24 = 2\pi r \]
To isolate \(r\), we can divide both sides of the equation by \(2\pi\):
\[ r = \frac{24}{2\pi} = \frac{12}{\pi} \]
Now, we will calculate the value of \(r\) using the value of \(\pi \approx 3.14159\):
\[ r \approx \frac{12}{3.14159} \approx 3.8197 \]
Rounding this value to the nearest hundredth:
\[ r \approx 3.82 \]
Thus, the radius of the lake is approximately:
\[ \boxed{3.82} \text{ miles} \]