Answer :
To determine how much farther Addison can see to the horizon compared to Kaylib, we'll use the formula for the distance to the horizon, given by [tex]\( d = \sqrt{\frac{3h}{2}} \)[/tex], where [tex]\( d \)[/tex] is the distance to the horizon in miles and [tex]\( h \)[/tex] is the height above sea level in feet.
### Step-by-Step Solution:
1. Determine Kaylib's Distance to the Horizon:
- Kaylib's eye-level height [tex]\( h_{\text{Kaylib}} \)[/tex] is [tex]\( 48 \)[/tex] feet.
- Using the formula [tex]\( d = \sqrt{\frac{3h}{2}} \)[/tex]:
[tex]\[ d_{\text{Kaylib}} = \sqrt{\frac{3 \times 48}{2}} \][/tex]
[tex]\[ d_{\text{Kaylib}} = \sqrt{\frac{144}{2}} \][/tex]
[tex]\[ d_{\text{Kaylib}} = \sqrt{72} \][/tex]
[tex]\[ d_{\text{Kaylib}} \approx 8.485 \text{ miles} \][/tex]
2. Determine Addison's Distance to the Horizon:
- Addison's eye-level height [tex]\( h_{\text{Addison}} \)[/tex] is [tex]\( 85 \frac{1}{3} \approx 85.333 \)[/tex] feet.
- Using the formula [tex]\( d = \sqrt{\frac{3h}{2}} \)[/tex]:
[tex]\[ d_{\text{Addison}} = \sqrt{\frac{3 \times 85.333}{2}} \][/tex]
[tex]\[ d_{\text{Addison}} = \sqrt{\frac{256}{2}} \][/tex]
[tex]\[ d_{\text{Addison}} = \sqrt{128} \][/tex]
[tex]\[ d_{\text{Addison}} \approx 11.314 \text{ miles} \][/tex]
3. Calculate the Difference in Their Distances to the Horizon:
To find how much farther Addison can see compared to Kaylib, subtract the distance Kaylib can see from the distance Addison can see:
[tex]\[ \text{Difference} = d_{\text{Addison}} - d_{\text{Kaylib}} \][/tex]
[tex]\[ \text{Difference} \approx 11.314 - 8.485 \][/tex]
[tex]\[ \text{Difference} \approx 2.829 \text{ miles} \][/tex]
Therefore, Addison can see approximately [tex]\( 2.8284271247461916 \)[/tex] miles farther than Kaylib.
Among the provided options, the correct one that matches this value is [tex]\( 2 \sqrt{2} \)[/tex] miles, noting that [tex]\( 2\sqrt{2} \)[/tex] is approximately [tex]\( 2.828 \)[/tex].
Answer: [tex]\( 2 \sqrt{2} \)[/tex] miles
### Step-by-Step Solution:
1. Determine Kaylib's Distance to the Horizon:
- Kaylib's eye-level height [tex]\( h_{\text{Kaylib}} \)[/tex] is [tex]\( 48 \)[/tex] feet.
- Using the formula [tex]\( d = \sqrt{\frac{3h}{2}} \)[/tex]:
[tex]\[ d_{\text{Kaylib}} = \sqrt{\frac{3 \times 48}{2}} \][/tex]
[tex]\[ d_{\text{Kaylib}} = \sqrt{\frac{144}{2}} \][/tex]
[tex]\[ d_{\text{Kaylib}} = \sqrt{72} \][/tex]
[tex]\[ d_{\text{Kaylib}} \approx 8.485 \text{ miles} \][/tex]
2. Determine Addison's Distance to the Horizon:
- Addison's eye-level height [tex]\( h_{\text{Addison}} \)[/tex] is [tex]\( 85 \frac{1}{3} \approx 85.333 \)[/tex] feet.
- Using the formula [tex]\( d = \sqrt{\frac{3h}{2}} \)[/tex]:
[tex]\[ d_{\text{Addison}} = \sqrt{\frac{3 \times 85.333}{2}} \][/tex]
[tex]\[ d_{\text{Addison}} = \sqrt{\frac{256}{2}} \][/tex]
[tex]\[ d_{\text{Addison}} = \sqrt{128} \][/tex]
[tex]\[ d_{\text{Addison}} \approx 11.314 \text{ miles} \][/tex]
3. Calculate the Difference in Their Distances to the Horizon:
To find how much farther Addison can see compared to Kaylib, subtract the distance Kaylib can see from the distance Addison can see:
[tex]\[ \text{Difference} = d_{\text{Addison}} - d_{\text{Kaylib}} \][/tex]
[tex]\[ \text{Difference} \approx 11.314 - 8.485 \][/tex]
[tex]\[ \text{Difference} \approx 2.829 \text{ miles} \][/tex]
Therefore, Addison can see approximately [tex]\( 2.8284271247461916 \)[/tex] miles farther than Kaylib.
Among the provided options, the correct one that matches this value is [tex]\( 2 \sqrt{2} \)[/tex] miles, noting that [tex]\( 2\sqrt{2} \)[/tex] is approximately [tex]\( 2.828 \)[/tex].
Answer: [tex]\( 2 \sqrt{2} \)[/tex] miles
