Problem 73 - Precalculus Mathematics for Calculus

Question

Find the distance that the earth travels in one day in its path around the sun. Assume that a year has 365 days and that the path of the earth around the sun is a circle of radius 93 million miles. [The path of the earth around the sun is actually an ellipse with the sun at one focus (see Section 10.2 ). This ellipse, however, has very small eccentricity, so it is nearly circular. \(]\) (IMAGE CAN NOT COPY)

Step-by-Step Solution

Answer

Earth travels approximately 1,600,109 miles each day around the Sun.

Step 1: Calculate the Circumference of Earth's Orbit

The path of the Earth around the Sun is approximated as a circle with a radius of 93 million miles. To find the circumference, use the formula \( C = 2 \pi r \), where \( r \) is the radius. Substituting the given radius, we get:\[C = 2 \pi \times 93,000,000 \text{ miles}.\]

Step 2: Simplify the Circumference Calculation

Now, calculate the circumference using the approximation \( \pi \approx 3.1416 \):\[C = 2 \times 3.1416 \times 93,000,000 \approx 584,040,000 \text{ miles}.\]So, the circumference of Earth's orbit is approximately 584,040,000 miles.

Step 3: Determine the Distance Traveled in One Day

Since the Earth travels around the Sun once in 365 days, divide the circumference by 365 to find the daily travel distance:\[\text{Distance per day} = \frac{584,040,000 \text{ miles}}{365} \approx 1,600,109 \text{ miles}.\]

Key Concepts

Earth's orbit

The Earth's path around the Sun is fascinating when we think about it as more than just a year passing between birthdays. This path is generally understood as a circle, although it's technically a very slightly squashed circle known as an ellipse. This almost-circular path is essential for understanding the distances involved in the Earth's journey. To find out how far the Earth travels in its entire orbit, we measure the circumference of this circle-like path.
Using a radius of about 93 million miles - which is the average distance from the Earth to the Sun - we can calculate this circumference to be approximately 584 million miles. This gives us a true sense of the vast space that the Earth covers during a year.

• The Earth's orbit is nearly circular.
• Radius of orbit: 93 million miles.
• Calculating the full path (circumference) gives us an idea about the sheer scale of the Earth's journey around the Sun.

Circular motion

The concept of circular motion helps in understanding how the Earth moves around the Sun. Although the real path is slightly elliptical, using the notion of a circular path is a helpful simplification. This method allows us to apply the formula for the circumference of a circle, which is a crucial step to derive the daily travel distance of the Earth.
In circular motion, all parts of the circle are evenly traveled, based on one key measurement: the radius. With Earth's orbit, we use the radius of 93 million miles and apply it in the formula for circumference, which is \( C = 2 \pi r \).

• Circular motion simplifies complex calculations.
• It uses a fixed radius to calculate the distance traveled in a complete orbit.
• This is a foundational method for finding out how far Earth moves daily.

Day calculations

To understand how far the Earth travels in one day, we take the entire yearly journey and break it down into daily pieces. Once we know the circumference of Earth's path, the next logical step is to figure out how much of that path is completed each day. Since a year is about 365 days, we simply divide the total circumference by this number.
This means the Earth travels approximately 1,600,109 miles in one day, showcasing how even a single day covers an amazing distance on its path around the Sun. This division not only gives us a daily estimate but also helps us appreciate the continuous motion Earth maintains.

• The total path is divided by 365 to find the daily distance.
• One day covers over a million miles!
• This calculation emphasizes the constant and rapid pace of Earth's orbit.