To determine the length of the horizontal axis of the ellipse, we need to apply the geometric properties of ellipses.
The general equation governing the relationship between the semi-major axis (denoted as [tex]\( a \)[/tex]), the semi-minor axis (denoted as [tex]\( b \)[/tex]), and the distance between the foci (denoted as [tex]\( c \)[/tex]) is [tex]\( c^2 = a^2 - b^2 \)[/tex].
Here are the steps we need to follow:
1. Identify the given values:
- The total distance between the foci is 60 feet, so [tex]\( 2c = 60 \)[/tex] feet, which means [tex]\( c = 30 \)[/tex] feet.
- The length of the vertical axis is 200 feet, so the semi-major axis [tex]\( b \)[/tex] is [tex]\( \frac{200}{2} = 100 \)[/tex] feet.
2. Use the relationship [tex]\( c^2 = a^2 - b^2 \)[/tex] to find the semi-minor axis [tex]\( a \)[/tex]:
- Substituting the known values, we get [tex]\( 30^2 = a^2 - 100^2 \)[/tex].
- This simplifies to [tex]\( 900 = a^2 - 10000 \)[/tex].
- Solving for [tex]\( a^2 \)[/tex], we obtain [tex]\( a^2 = 900 + 10000 = 10900 \)[/tex].
3. Calculate [tex]\( a \)[/tex] (semi-minor axis):
- [tex]\( a = \sqrt{10900} \approx 104.4030650891055 \)[/tex] feet.
4. Determine the total length of the horizontal axis:
- The horizontal axis is twice the length of the semi-minor axis, so it is [tex]\( 2a \)[/tex].
5. Therefore, the length of the horizontal axis is:
- [tex]\( 2 \times 104.4030650891055 \approx 208.806130178211 \)[/tex] feet.
Thus, the horizontal axis of the ellipse-shaped resort is approximately 208.81 feet.
Therefore, the correct answer to select from the drop-down menu is 208.81 feet.
