To determine how much farther Adam can see to the horizon compared to Pam, we need to use the formula for the distance to the horizon given an observer's height. The formula is:
[tex]\[ d = \sqrt{\frac{3h}{2}} \][/tex]
where:
- [tex]\( d \)[/tex] is the distance to the horizon in miles
- [tex]\( h \)[/tex] is the eye-level height above sea level in feet
First, let's find the distance to the horizon for Adam, whose eye-level height is 400 feet.
[tex]\[ d_{Adam} = \sqrt{\frac{3 \times 400}{2}} \][/tex]
[tex]\[ d_{Adam} = \sqrt{\frac{1200}{2}} \][/tex]
[tex]\[ d_{Adam} = \sqrt{600} \][/tex]
Next, let's find the distance to the horizon for Pam, whose eye-level height is 256 feet.
[tex]\[ d_{Pam} = \sqrt{\frac{3 \times 256}{2}} \][/tex]
[tex]\[ d_{Pam} = \sqrt{\frac{768}{2}} \][/tex]
[tex]\[ d_{Pam} = \sqrt{384} \][/tex]
Now, to determine how much farther Adam can see compared to Pam, we need to find the difference between Adam's and Pam's distances to the horizon.
The correct expression showing how much farther Adam can see is:
[tex]\[ \sqrt{\frac{3 \times 400}{2}} - \sqrt{\frac{3 \times 256}{2}} \][/tex]
Using the values we calculated:
[tex]\[ d_{Adam} = 24.49489742783178 \][/tex]
[tex]\[ d_{Pam} = 19.595917942265423 \][/tex]
The difference is:
[tex]\[ 24.49489742783178 - 19.595917942265423 = 4.8989794855663575 \][/tex]
Therefore, the correct answer is the expression:
[tex]\[ \sqrt{\frac{3(400)}{2}} - \sqrt{\frac{3(256)}{2}} \][/tex]
