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Chapter 1: Problem 20

A person is standing at the edge of the water and looking out at the ocean (see the drawing). The height of the person's eyes above the water is \(h=1.6 \mathrm{m},\) and the radius of the earth is \(R=6.38 \times 10^{6} \mathrm{m}\) (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 \(90^{\circ} .\) ) (b) Express this distance in miles.

Short Answer

Expert verified
Distance to the horizon is approximately 4517.9 meters or 2.81 miles.

Step by step solution

01

Understand the problem

You need to find the distance \(d\) from the person's eyes to the horizon, where the eyes are at a height \(h = 1.6 \text{ m}\) above the water. The radius of the Earth is \(R = 6.38 \times 10^{6} \text{ m}\). The line of sight at the horizon makes a \(90^\circ\) angle with the Earth's radius.
02

Use the right triangle relationship

The right triangle formed includes the Earth's radius \(R\), the radius plus the height \(R + h\), and the distance \(d\) as the hypotenuse. Therefore, according to the Pythagorean theorem: \( (R + h)^2 = R^2 + d^2 \).
03

Solve for \(d\)

Rearrange the equation \((R + h)^2 = R^2 + d^2 \) to solve for \(d\). We have:\[ d^2 = (R + h)^2 - R^2 = R^2 + 2Rh + h^2 - R^2 = 2Rh + h^2 \]\[ d = \sqrt{2Rh + h^2} \]
04

Substitute values to find \(d\)

Substitute \(R = 6.38 \times 10^6 \text{ m}\) and \(h = 1.6 \text{ m}\) into the equation: \[ d = \sqrt{2(6.38 \times 10^6)(1.6) + (1.6)^2} \] Calculate \(d\) to find the distance to the horizon in meters.
05

Calculate \(d\) in meters

Plug in the values to find \(d:\)\[ d = \sqrt{2 \times 6.38 \times 10^6 \times 1.6 + 1.6^2} \approx \sqrt{20,416,000 + 2.56} \approx \sqrt{20,416,002.56} \approx 4,517.9 \text{ m} \]
06

Convert distance from meters to miles

Use the conversion factor \(1 \text{ mile} = 1609.34 \text{ meters}\). The distance \(d\) in miles is given by: \[ d_{\text{miles}} = \frac{4517.9}{1609.34} \approx 2.81 \text{ miles} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pythagorean Theorem
To find the distance to the horizon, understanding the Pythagorean theorem is essential. The Pythagorean theorem helps us solve problems involving right-angled triangles. Essentially, it states that in a right triangle, the square of the hypotenuse's length is equal to the sum of the squares of the other two sides:
  • The formula is: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.
  • In this problem, the hypotenuse \( c \) is the distance \( d \) to the horizon.
  • The other two sides are the earth's radius \( R \) and the combined radius plus height \( R + h \).
Using this theorem, you can determine the distance \( d \) by rearranging the equation: \( d^2 = (R + h)^2 - R^2 \). It cleverly accommodates the height of the person standing on the surface of the Earth, seeing how it affects the visible horizon.
Earth's Radius
Earth's radius plays a significant role in calculating the visible horizon. Understanding that the Earth is largely spherical helps us apply geometric principles to real-world scenarios. On average, the Earth's radius is about \( R = 6.38 \times 10^6 \text{ m} \).
This constant is critical in many physics and engineering calculations, impacting how distances are perceived from different heights.
  • Since the person's eyes are a small height \( h \) above the surface, it slightly alters the total radius.
  • The total radius with the height included becomes \( R + h \).
This modified radius aids in effectively using the Pythagorean theorem to find the line of sight horizon distance. Understanding Earth's size helps frame how these everyday calculations work for something as simple as determining how far we see over the ocean.
Unit Conversion
Converting units is integral to translating calculations into practical results. In this problem, you initially find the distance to the horizon in meters. Yet, it's useful to convert this distance into miles, a unit more commonly used in certain regions or contexts.
To convert meters to miles:
  • We use the conversion factor: \( 1 ext{ mile} = 1609.34 ext{ meters} \).
  • Thus, to convert the calculated distance \( 4517.9 \text{ meters} \) to miles, you divide by the number of meters in a mile.
This simple division facilitates easy interpretation of distances for everyday applications, providing a sense of scale that feels more relatable depending on cultural contexts.
Trigonometry
Trigonometry provides essential tools for understanding how angles and lengths relate within geometric figures like triangles. In this particular problem, the angle between the line of sight to the horizon and Earth's radius is \(90^{\circ}\), meaning we are dealing with a right triangle.
  • Trigonometry helps decipher relationships between the sides and angles of right triangles.
  • It uses angles to simplify the understanding of geometrical concepts that arise naturally when observing phenomena like horizons from varying heights.
Since the problem revolves around a right triangle established by the vertical height, the Earth's radius, and the hypotenuse as the line of sight, trigonometric principles simplify understanding distance calculations and the perception of how variables like height affect visibility over the Earth's curvature.

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