Let's break down the problem step-by-step using the given formula and information.
### Step 1: Determine Heights
- Kaylib's height [tex]\( h_k \)[/tex] is 48 feet.
- Addison's height [tex]\( h_a \)[/tex] is [tex]\( 85 + \frac{1}{3} \)[/tex] feet. This is a mixed number which we can convert to an improper fraction or decimal:
[tex]\[
85 + \frac{1}{3} = 85 + 0.333\overline{3} \approx 85.3333 \text{ feet}
\][/tex]
### Step 2: Calculate the Distance to the Horizon for Each Person
We use the formula:
[tex]\[
d = \sqrt{\frac{3h}{2}}
\][/tex]
- For Kaylib:
[tex]\[
d_k = \sqrt{\frac{3 \cdot 48}{2}}
\][/tex]
First, compute the inner part:
[tex]\[
\frac{3 \cdot 48}{2} = \frac{144}{2} = 72
\][/tex]
Then, compute the square root:
[tex]\[
d_k = \sqrt{72}
\][/tex]
Using the value obtained from the data:
[tex]\[
d_k \approx 8.4853 \text{ miles}
\][/tex]
- For Addison:
[tex]\[
d_a = \sqrt{\frac{3 \cdot 85.3333}{2}}
\][/tex]
First, compute the inner part:
[tex]\[
\frac{3 \cdot 85.3333}{2} = \frac{256.0}{2} = 128.0
\][/tex]
Then, compute the square root:
[tex]\[
d_a = \sqrt{128.0}
\][/tex]
Using the value obtained from the data:
[tex]\[
d_a \approx 11.3137 \text{ miles}
\][/tex]
### Step 3: Calculate the Difference
To find how much farther Addison can see compared to Kaylib, subtract Kaylib's distance from Addison's distance:
[tex]\[
\text{Difference} = d_a - d_k
\][/tex]
Using the values:
[tex]\[
\text{Difference} = 11.3137 - 8.4853 = 2.8284 \text{ miles}
\][/tex]
### Step 4: Match to Given Options
From the options provided, we recognize that:
[tex]\[
2.8284 \approx 2 \sqrt{2}
\][/tex]
since [tex]\(\sqrt{2} \approx 1.414\)[/tex], so:
[tex]\[
2 \cdot \sqrt{2} \approx 2 \cdot 1.414 = 2.828
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{2 \sqrt{2} \text{ miles}}
\][/tex]
