To find the length of segment \( LF \), we can use some properties of circles and the relationship between tangents and angles.
Given:
- \( KM \) and \( KL \) are tangents to the circle.
- \( KM = 47 , \text{ft} \).
- \( \angle LFK = 23^\circ \).
From the properties of tangents to a circle:
- The angle between a tangent and a radius drawn to the point of contact is \( 90^\circ \).
- The tangents \( KM \) and \( KL \) are equal in length because they are tangents from the same external point \( K \).
Thus, we have:
- \( KL = 47 , \text{ft} \).
Now, consider triangle \( KLF \):
- Since both \( KM \) and \( KL \) are tangents, there is a right triangle formed with the radius at the point of tangency to side \( LF \) (which connects \( L \) and \( F \)), meaning \( \angle KLF = 90^\circ \).
To determine \( LF \), we can apply the tangent function in triangle \( KLF \):
- We know \( \angle LFK = 23^\circ \) and that the opposite side to \( \angle LFK \) is \( LF \) and the adjacent side is \( KL \).
Using the tangent function: \[ \tan(\angle LFK) = \frac{LF}{KL} \] Substituting the known values: \[ \tan(23^\circ) = \frac{LF}{47} \]
Now, calculate \( \tan(23^\circ) \): \[ \tan(23^\circ) \approx 0.4245 \]
We then substitute this back into our equation: \[ 0.4245 = \frac{LF}{47} \]
Solving for \( LF \): \[ LF = 47 \times 0.4245 \approx 19.94 \]
Rounded to the nearest foot, \( LF \) is: \[ \text{Length of } LF \approx 20 \text{ ft}. \]
Thus, the length of \( LF \) to the nearest foot is \( \boxed{20} \).