To determine the distance to the horizon from a viewpoint that is 10 miles above the Earth's surface, you can use the following formula for the distance to the horizon:
[tex]\[ \text{distance} = \sqrt{2 \cdot \text{radius} \cdot \text{height} + \text{height}^2} \][/tex]
Here's how to solve it step-by-step:
1. Identify the given values:
- Radius of the Earth ([tex]\( r \)[/tex]) = 3959 miles
- Height above the Earth's surface ([tex]\( h \)[/tex]) = 10 miles
2. Substitute these values into the formula:
[tex]\[ \text{distance} = \sqrt{2 \cdot 3959 \cdot 10 + 10^2} \][/tex]
3. Break it down inside the sqrt calculation:
- First calculate [tex]\( 2 \cdot 3959 \cdot 10 \)[/tex]:
[tex]\( 2 \cdot 3959 \cdot 10 = 79180 \)[/tex]
- Then calculate [tex]\( 10^2 \)[/tex]:
[tex]\( 10^2 = 100 \)[/tex]
- Now add these two results together:
[tex]\( 79180 + 100 = 79280 \)[/tex]
4. Take the square root of the sum:
[tex]\[ \sqrt{79280} \approx 281.5670435260491 \][/tex]
5. Finally, round the result to the nearest mile:
[tex]\[ \approx 282 \text{ miles} \][/tex]
So, the distance to the horizon from a viewpoint that is 10 miles above the Earth's surface is approximately 282 miles.
