To determine the distance each person can see to the horizon given their respective eye-level heights, we use the formula:
[tex]\[ d = \sqrt{\frac{3h}{2}} \][/tex]
Step-by-Step Solution:
1. Calculate Wyatt's distance to the horizon:
- Wyatt's eye-level height is [tex]\( h_{\text{wyatt}} = 120 \, \text{ft} \)[/tex].
- Plugging [tex]\( h_{\text{wyatt}} \)[/tex] into the formula, we get:
[tex]\[
d_{\text{wyatt}} = \sqrt{\frac{3 \times 120}{2}} = \sqrt{180} \approx 13.416407864998739 \, \text{miles}
\][/tex]
2. Calculate Shawn's distance to the horizon:
- Shawn's eye-level height is [tex]\( h_{\text{shawn}} = 270 \, \text{ft} \)[/tex].
- Plugging [tex]\( h_{\text{shawn}} \)[/tex] into the formula, we get:
[tex]\[
d_{\text{shawn}} = \sqrt{\frac{3 \times 270}{2}} = \sqrt{405} \approx 20.12461179749811 \, \text{miles}
\][/tex]
3. Find the difference in distances to the horizon:
- Subtract Wyatt's distance from Shawn's distance:
[tex]\[
\text{distance difference} = d_{\text{shawn}} - d_{\text{wyatt}} \approx 20.12461179749811 - 13.416407864998739 \approx 6.708203932499369 \, \text{miles}
\][/tex]
Thus, the answer to how much farther Shawn can see compared to Wyatt is [tex]\( \boxed{6.708203932499369} \)[/tex] miles.
Since none of the provided answer choices match this number exactly in the form they are presented (as forms involving [tex]\(\sqrt{5}\)[/tex]), this suggests that none of the given multiple-choice answers are correct within the context of this calculation. However, if the options given should represent the calculated difference in another form, it should be clarified that these do not align with the accurate numerical approximation we have derived [tex]\( 6.708203932499369 \)[/tex].
