To determine how much farther Addison can see to the horizon than Kaylib, let's follow the given formula [tex]\(d = \sqrt{\frac{3h}{2}}\)[/tex], where [tex]\(d\)[/tex] represents the distance to the horizon in miles and [tex]\(h\)[/tex] represents the eye-level height in feet.
First, let's calculate the distance for each of them individually.
### Kaylib's Distance to the Horizon:
Kaylib's eye-level height [tex]\(h_kaylib\)[/tex] is 48 feet.
Using the formula [tex]\(d = \sqrt{\frac{3h}{2}}\)[/tex], we get:
[tex]\[ d_{kaylib} = \sqrt{\frac{3 \cdot 48}{2}} \][/tex]
[tex]\[ d_{kaylib} = \sqrt{\frac{144}{2}} \][/tex]
[tex]\[ d_{kaylib} = \sqrt{72} \][/tex]
[tex]\[ d_{kaylib} = 8.485 \text{ miles} \][/tex]
### Addison's Distance to the Horizon:
Addison's eye-level height [tex]\(h_addison\)[/tex] is [tex]\(85 \frac{1}{3}\)[/tex] feet, which is equal to [tex]\(\frac{256}{3}\)[/tex] feet.
Using the formula [tex]\(d = \sqrt{\frac{3h}{2}}\)[/tex], we get:
[tex]\[ d_{addison} = \sqrt{\frac{3 \cdot 85 \frac{1}{3}}{2}} \][/tex]
[tex]\[ d_{addison} = \sqrt{\frac{3 \cdot 256/3}{2}} \][/tex]
[tex]\[ d_{addison} = \sqrt{\frac{256}{2}} \][/tex]
[tex]\[ d_{addison} = \sqrt{128} \][/tex]
[tex]\[ d_{addison} = 11.314 \text{ miles} \][/tex]
### Difference in Distance:
Now, the difference between the distances they can see to the horizon is:
[tex]\[ \text{Difference} = d_{addison} - d_{kaylib} \][/tex]
[tex]\[ \text{Difference} = 11.314 - 8.485 \][/tex]
[tex]\[ \text{Difference} = 2.829 \text{ miles} \][/tex]
### Simplified to Options:
Given the list of options:
- [tex]\(\sqrt{2}\)[/tex] miles
- [tex]\(2\sqrt{2}\)[/tex] miles
- [tex]\(14\sqrt{2}\)[/tex] miles
- [tex]\(28\sqrt{2}\)[/tex] miles
We need to express [tex]\(2.829\)[/tex] miles in terms of [tex]\(\sqrt{2}\)[/tex]. We know:
[tex]\[ \sqrt{2} \approx 1.414 \][/tex]
So,
[tex]\[ 2 \sqrt{2} \approx 2 \times 1.414 = 2.828 \][/tex]
Thus, [tex]\(2.828 \approx 2.829\)[/tex], implying:
[tex]\[ 2.829 \approx 2\sqrt{2} \][/tex]
Hence, Addison can see approximately [tex]\(2 \sqrt{2}\)[/tex] miles farther than Kaylib. Therefore, the correct answer is:
[tex]\[
\boxed{2\sqrt{2} \text{ miles}}
\][/tex]
