A person standing h feet above sea level can seecd miles to the horizon. The distance r is the radius of the earth = 3963 miles. d is the tangent to the circle forming a right angle to the radius. h + r is the tangent. Solve using Pythagorean theorem and secant-tangent theorem. I don't know if this is important but the question also says mount everest summit reaches 29,035 feet above sea level.

Question

A person standing h feet above sea level can seecd miles to the horizon. The distance r is the radius of the earth = 3963 miles. d is the tangent to the circle forming a right angle to the radius. h + r is the tangent. Solve using Pythagorean theorem and secant-tangent theorem. I don't know if this is important but the question also says mount everest summit reaches 29,035 feet above sea level.

Answer 1

If you draw a triangle where the hypotenuse is r+h (the elevated observation point), and the distance to the horizon is d, then

r^2+d^2 = (r+h)^2

Answer 2

Thanks Steve. I got it. Forgot to convert feet to miles when calculating.

Answer 3

To solve this problem using the Pythagorean theorem and the secant-tangent theorem, we can follow these steps:

Step 1: Understand the given information.
- We are given that a person standing h feet above sea level can see a distance of d miles to the horizon.
- The distance r is the radius of the Earth, which is 3963 miles.

Step 2: Formulate the problem.
- We need to determine the relationship between h, d, and r using the Pythagorean theorem and the secant-tangent theorem.

Step 3: Use the Pythagorean theorem.
- The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
- In our case, h + r is the hypotenuse, and d is one of the legs.

Applying the Pythagorean theorem:
(h + r)^2 = d^2 + r^2

Step 4: Simplify the equation.
Expanding the equation:
h^2 + 2hr + r^2 = d^2 + r^2

Simplifying the equation:
h^2 + 2hr = d^2

Step 5: Solve for d.
We were given the height of Mount Everest, which is h = 29,035 feet above sea level. We need to calculate d, the distance to the horizon, using the given height and radius of the Earth.

Substituting the values into the equation:
(29,035 + 3963)^2 = d^2

Solving for d:
(33,998)^2 = d^2
d^2 = 1,156,403,604
d ≈ 34,021 miles

Therefore, the person standing at a height of 29,035 feet above sea level can see approximately 34,021 miles to the horizon.

Note: The given values for h and r in this explanation are specific to Mount Everest. If you have different values for h or r, you can substitute those values into the equation to calculate the distance to the horizon for any other height above sea level.