- Identify the heights for Kaylib \(h_K = 48\) ft and Addison \(h_A = \frac{256}{3}\) ft.
- Apply the formula \(d = \sqrt{\frac{3h}{2}}\) to find their respective visibility distances: \(d_K = 6\sqrt{2}\) mi and \(d_A = 8\sqrt{2}\) mi.
- Calculate the difference between their distances: \(\Delta d = 8\sqrt{2} - 6\sqrt{2} = 2\sqrt{2}\) mi.
- The final answer is \(\boxed{2\sqrt{2}}\) mi.
Explanation
- Understanding the Problem
We are given the eye-level heights of Kaylib and Addison, \(h_K = 48\) ft and \(h_A = 85 \frac{1}{3} = \frac{256}{3}\) ft respectively. We are also given a formula for the distance to the horizon \(d = \sqrt{\frac{3h}{2}}\), where \(d\) is in miles and \(h\) is in feet. Our objective is to find the difference in the distance that Addison and Kaylib can see to the horizon, which is \(\Delta d = d_A - d_K\).
- Calculating Kaylib's Distance to the Horizon
First, we calculate the distance Kaylib can see to the horizon by substituting \(h_K = 48\) into the formula:
\[d_K = \sqrt{\frac{3 \times 48}{2}} = \sqrt{\frac{144}{2}} = \sqrt{72}.\]
Simplifying the radical, we have \(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\) miles.
- Calculating Addison's Distance to the Horizon
Next, we calculate the distance Addison can see to the horizon by substituting \(h_A = \frac{256}{3}\) into the formula:
\[d_A = \sqrt{\frac{3 \times \frac{256}{3}}{2}} = \sqrt{\frac{256}{2}} = \sqrt{128}.\]
Simplifying the radical, we have \(\sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}\) miles.
- Calculating the Difference and Conclusion
Finally, we find the difference in the distances that Addison and Kaylib can see:
\[\Delta d = d_A - d_K = 8\sqrt{2} - 6\sqrt{2} = 2\sqrt{2}.\]
Thus, Addison can see \(2\sqrt{2}\) miles farther than Kaylib. Comparing this result with the given options, we select the option \(2\sqrt{2}\) mi. The final answer is \(\boxed{2\sqrt{2}}\) mi.
Examples
This problem illustrates how physical measurements (like height) relate to geometric outcomes (like the distance to the horizon). In real life, sailors or lighthouse keepers use similar formulas to determine how far out at sea they can spot objects or ships. By understanding these relationships, one can estimate visibility ranges based on altitude, which is essential for navigation, safety, and surveillance. Such calculations are foundational in fields like meteorology and maritime operations.