Question

The summit of Mount Mauna Kea is approximately 13,796 ft above sea level. What is the distance from the summit to the horizon, rounded to the nearest mile? Assume that the distance from Earth's center to any point on Earth's surface is 4,000 miles.

Answer 1

≈ 145 mi

Explanation:

Answer 2

distance from the summit to the horizon = 145 miles

Explanation:

I have attached a diagram to depict the question;

We are given that the distance from Earth's center to any point on Earth's surface is 4,000 .

Thus, the value of r in the diagram is; r = 4000 miles

Now we are also told that the summit of Mount Mauna Kea is approximately 13,796 ft above sea level. Thus, v in the diagram is v = 13,796 ft.

Since r is in miles, let's convert it to miles too.

Thus, v = 13796 x (1/5280) = 2.612879 miles

Thus, v + r = 2.612879 miles + 4000 miles = 4002.612879 miles

we are trying to find the distance from the summit to the horizon. In the diagram, it is represented by the side OH.

Thus, using Pythagoras theorem, we have;

(OH)² + r² = (v + r)²

Thus,plugging in the values and rearranging;

(OH)² = 4002.612879² - 4000²

(OH)² = √20909.85913666699

OH = 144.6 miles ≈ 145 miles