A person is standing at the edge of the water and looking out at the ocean (see figure). The height of the person's eyes above the water is h = 1.9 m, and the radius of the Earth is R = 6.38 x 106 m. (a) How far is it to the horizon? In other words, what is the distance d from the person's eyes to the horizon? (Note: At the horizon the angle between the line of sight and the radius of the earth is 90°.) (b) Express this distance in miles.
To calculate the distance from the person's eyes to the horizon, we can use the Pythagorean theorem. Let's denote the distance as d.
(a) Using the Pythagorean theorem, we have:
d^2 = R^2 + (R+h)^2
where R is the radius of the Earth and h is the height of the person's eyes above the water. Plugging in the values, we get:
d^2 = (6.38 x 10^6)^2 + (6.38 x 10^6 + 1.9)^2
Calculating this expression, we find:
d^2 ≈ 2.739 x 10^13
Taking the square root of both sides, we find:
d ≈ 1.655 x 10^7 m
Therefore, the distance from the person's eyes to the horizon is approximately 1.655 x 10^7 meters.
(b) To express this distance in miles, we can use the conversion factor 1 mile = 1609.34 meters. So, we have:
d_miles = d / 1609.34
Plugging in the value of d, we get:
d_miles ≈ (1.655 x 10^7) / 1609.34
Calculating this expression, we find:
d_miles ≈ 10292 miles (rounded to the nearest mile)
Therefore, the distance from the person's eyes to the horizon is approximately 10292 miles.
To find the distance from a person's eyes to the horizon, we can use basic trigonometry. Let's break it down step by step.
(a) Calculation in meters:
We can form a right triangle using the person's eyes, the surface of the Earth, and the line of sight to the horizon. The radius of the Earth, R, is the length of the hypotenuse, and the distance from the eyes to the horizon, d, is the length of one of the legs.
Using Pythagoras' theorem, we have:
R^2 = d^2 + h^2
Substituting the given values:
(6.38 x 10^6)^2 = d^2 + (1.9)^2
Simplifying the equation:
(4.07 x 10^13) = d^2 + 3.61
Rearranging the equation to solve for d:
d^2 = (4.07 x 10^13) - 3.61
d^2 = (4.07 x 10^13)
Taking the square root of both sides to find d:
d ≈ 6.37 x 10^6 meters
Therefore, the distance from the person's eyes to the horizon is approximately 6.37 x 10^6 meters.
(b) Conversion to miles:
To express this distance in miles, we need to convert meters to miles. Since 1 mile is approximately 1609 meters, we can use this conversion factor.
d (in miles) = d (in meters) / 1609
Substituting the calculated value:
d (in miles) ≈ (6.37 x 10^6) / 1609
Simplifying the expression:
d (in miles) ≈ 3,958 miles
Therefore, the distance from the person's eyes to the horizon is approximately 3,958 miles.
21
4500m
2.8miles