Answer :
To find the distance from the top of Mt. Everest to the Earth's horizon, we can use a geometric approach based on the properties of circles.
Here’s a step-by-step guide to solving this problem:
1. Understand the Scenario: Imagine a right triangle where:
- One vertex is at the summit of Mt. Everest.
- Another vertex is at the point on Earth's surface directly below the summit.
- The last vertex is the point where the horizon is visible from the summit.
2. Given Values:
- The radius of the Earth (R) is 4000 miles.
- The height from Earth's surface to the summit of Mt. Everest (h) is 5.5 miles.
3. Formula for Distance to Horizon:
The distance (d) from the top of a height to the horizon can be derived using the Pythagorean theorem and is given by the formula:
[tex]\[
d = \sqrt{h^2 + 2 \times R \times h}
\][/tex]
Where:
- [tex]\( h = 5.5 \)[/tex] miles (height of Mt. Everest)
- [tex]\( R = 4000 \)[/tex] miles (radius of the Earth)
4. Calculate the Distance:
Plug the values into the formula:
[tex]\[
d = \sqrt{(5.5)^2 + 2 \times 4000 \times 5.5}
\][/tex]
5. Simplify and Calculate:
- First, calculate [tex]\( 5.5^2 = 30.25 \)[/tex]
- Next, calculate [tex]\( 2 \times 4000 \times 5.5 = 44000 \)[/tex]
- Add these results: [tex]\( 30.25 + 44000 = 44030.25 \)[/tex]
- Finally, take the square root: [tex]\( \sqrt{44030.25} \approx 209.8 \)[/tex]
6. Round to the Nearest Tenth:
The distance to the horizon from the summit is approximately 209.8 miles when rounded to the nearest tenth.
So, to the nearest tenth of a mile, the distance from the summit of Mt. Everest to the Earth's horizon is 209.8 miles.
Here’s a step-by-step guide to solving this problem:
1. Understand the Scenario: Imagine a right triangle where:
- One vertex is at the summit of Mt. Everest.
- Another vertex is at the point on Earth's surface directly below the summit.
- The last vertex is the point where the horizon is visible from the summit.
2. Given Values:
- The radius of the Earth (R) is 4000 miles.
- The height from Earth's surface to the summit of Mt. Everest (h) is 5.5 miles.
3. Formula for Distance to Horizon:
The distance (d) from the top of a height to the horizon can be derived using the Pythagorean theorem and is given by the formula:
[tex]\[
d = \sqrt{h^2 + 2 \times R \times h}
\][/tex]
Where:
- [tex]\( h = 5.5 \)[/tex] miles (height of Mt. Everest)
- [tex]\( R = 4000 \)[/tex] miles (radius of the Earth)
4. Calculate the Distance:
Plug the values into the formula:
[tex]\[
d = \sqrt{(5.5)^2 + 2 \times 4000 \times 5.5}
\][/tex]
5. Simplify and Calculate:
- First, calculate [tex]\( 5.5^2 = 30.25 \)[/tex]
- Next, calculate [tex]\( 2 \times 4000 \times 5.5 = 44000 \)[/tex]
- Add these results: [tex]\( 30.25 + 44000 = 44030.25 \)[/tex]
- Finally, take the square root: [tex]\( \sqrt{44030.25} \approx 209.8 \)[/tex]
6. Round to the Nearest Tenth:
The distance to the horizon from the summit is approximately 209.8 miles when rounded to the nearest tenth.
So, to the nearest tenth of a mile, the distance from the summit of Mt. Everest to the Earth's horizon is 209.8 miles.
