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Chapter 2: Problem 41

Earth is spinning along at \(1,674 \mathrm{km} / \mathrm{h}\) at the equator. Use this fact, along with the length of the day, to calculate Earth's equatorial diameter.

Short Answer

Expert verified
Earth's equatorial diameter is approximately 12,792 km.

Step by step solution

01

Understand the Problem

The problem provides the speed of rotation at the equator and asks for the equatorial diameter. Use the relationship between speed, time, and distance to find the diameter.
02

Identify Given Values

The speed of Earth's rotation at the equator is given as: \[ v = 1,674 \text{ km/h} \] A day is 24 hours: \[ t = 24 \text{ hours} \]
03

Formula for Circumference

The distance Earth travels at the equator in one day is the circumference of the Earth. Use the formula: \[ c = v \times t \]
04

Calculate the Circumference

Substitute the given values into the formula: \[ c = 1,674 \text{ km/h} \times 24 \text{ hours} \] Calculate: \[ c = 40,176 \text{ km} \]
05

Diameter Relationship

Use the relationship between circumference and diameter: \[ c = \pi \times d \] Rearrange to solve for diameter: \[ d = \frac{c}{\pi} \]
06

Calculate the Diameter

Substitute the circumference value into the formula: \[ d = \frac{40,176 \text{ km}}{3.14159} \] Calculate: \[ d \approx 12,792 \text{ km} \]

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Key Concepts

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

Earth's rotation speed
The Earth spins at a remarkable speed at the equator. This rotation helps to determine many of its physical characteristics. Earth's rotation speed at the equator is given as 1,674 km/h. To visualize this, imagine a point on the equator traveling at this speed. This speed is constant, reflecting how fast the Earth rotates on its axis completing one full revolution in a day. Understanding this rotation speed is crucial for figuring out how far a point on the equator travels in one complete rotation, which ultimately helps in determining Earth's circumference.
Circumference formula
To determine the Earth's circumference, we use the basic formula for distance, given the speed and time of travel: \( c = v \times t \). Here, 'c' represents the circumference, 'v' is the rotational speed at the equator (1,674 km/h), and 't' is the time it takes for one full rotation (24 hours). By multiplying the speed by the time, you get the total distance that a point on the equator travels in a day. Therefore, Earth's circumference is found using the equation: \( c = 1,674 \text{ km/h} \times 24 \text{ hours} = 40,176 \text{ km} \). This calculation helps us find how extensive the circular path around Earth’s equator is.
Diameter calculation
Once we know the Earth's circumference, we can determine the diameter using the relationship between circumference and diameter. The formula connecting these two is: \( c = \pi \times d \), where \( \pi \) is a constant approximately equal to 3.14159 and 'd' represents the diameter. To find the diameter 'd', we rearrange the formula: \( d = \frac{c}{\pi} \). Using the circumference we calculated earlier (40,176 km), we substitute it into the equation: \( d = \frac{40,176 \text{ km}}{3.14159} \). Upon calculating, we find that the Earth's equatorial diameter is approximately 12,792 km.
Distance-speed-time relationship
Understanding the relationship between distance, speed, and time is essential for solving problems like this. The formula connecting these variables is: \( \text{Distance} = \text{Speed} \times \text{Time} \). In our problem, the circumference of Earth represents the distance traveled by a point on the equator in a day. The rotational speed gives us the rate at which this point moves, and time is the duration for one full revolution (24 hours). By applying the formula, we find: \( c = v \times t = 1,674 \text{ km/h} \times 24 \text{ hours} = 40,176 \text{ km} \). This straightforward relationship lets us convert between these units, making complex calculations more manageable.

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Most popular questions from this chapter

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