Problem 139 - A Graphical Approach to Precalculus with Limits

Question

The first accurate estimate of the distance around Earth was done by the Greek astronomer Eratosthenes \((276-195 \text { B.C.) },\) who noted that the noontime position of the sun at the summer solstice differed by \(7^{\circ} 12^{\prime}\) from the city of Syene to the city of Alexandria. (See the figure.) The distance between these two cities is 496 miles. Use the are length formula to estimate the radius of Earth. Then approximate the circumference of Earth.

Step-by-Step Solution

Answer

Estimated Earth's radius is about 3,960 miles and its circumference is approximately 24,840 miles.

Step 1: Understand Angle Difference

The angle difference given is between the sun at noon in two cities, Syene and Alexandria, which is noted to be \(7^{\circ}12'\). This angle can be converted entirely to degrees for simplicity in calculation. The conversion is as follows: \(7 + \frac{12}{60} = 7.2^{\circ}\).

Step 2: Apply Arc Length Formula

The arc length formula \(s = r \theta\) relates the arc length \(s\), the radius \(r\) of the circle, and the angle \(\theta\) in radians. We need to convert \(\theta = 7.2^{\circ}\) into radians, \(\theta = 7.2 \times \frac{\pi}{180} = \frac{7.2 \pi}{180}\).

Step 3: Calculate Earth's Radius

Given the distance \(s = 496\) miles is the arc length between the two cities, substitute into the formula: \(496 = r \cdot \frac{7.2 \pi}{180}\). Rearrange to find \(r\): \(r = \frac{496 \times 180}{7.2 \pi}\). Calculate \(r\).

Step 4: Calculate Earth's Circumference

The circumference \(C\) of a circle is given by \(C = 2\pi r\). Use the radius \(r\) obtained from Step 3 to compute the circumference: \(C = 2\pi \times \frac{496 \times 180}{7.2\pi}\). Simplify the expression to find the circumference of Earth.

Key Concepts

Arc Length Formula

The arc length formula is a crucial concept in geometry, especially when dealing with circles and circular objects. It helps in determining the length along a circular path between two points, using the radius of the circle and the angle subtended at the center by the arc. The formula is expressed as:

• \( s = r \theta \)

Here, \( s \) is the arc length, \( r \) represents the radius of the circle, and \( \theta \) is the angle in radians.
This formula shows that the arc length is directly proportional to both the radius and the angle, meaning larger angles and longer radii will result in longer arcs.
When applying this to Eratosthenes' calculation, the arc length corresponds to the distance between the cities of Syene and Alexandria, and the radius we are interested in determining is that of Earth.
Keep in mind the importance of using radians for the angle in this equation, as the relationship is defined based on radians.

Angle Conversion

Converting angles from degrees to radians is often necessary in mathematical formulas, particularly when they involve circular motions or geometry. The conversion is crucial because many formulas, such as the arc length formula, require angles in radians rather than degrees.
To convert from degrees to radians, you use the relationship:

• \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)

In the case of Eratosthenes' calculation:

• The angle between the sun at noon in Syene and Alexandria is given as \( 7^{\circ}12' \).
• This can be written as \(7.2^{\circ}\) by converting 12 minutes into a fractional degree: \( \frac{12}{60} = 0.2 \).
• Therefore, in radians, this angle is \(7.2 \times \frac{\pi}{180} \).

Radian conversion allows us to integrate this measurement into formulas that rely on radian measures, ensuring accuracy and consistency.

Circumference Calculation

Circumference calculation is essential for understanding the full extent of circular shapes and objects, such as Earth when approximating its size. The circumference is simply the total length around a circle when the line completely encloses it.
The formula to calculate the circumference \( C \) of a circle is:

• \( C = 2\pi r \)

Here, \( r \) denotes the radius of the circle. This formula shows how circumference directly scales with the radius.
In the context of Earth's circumference as calculated by Eratosthenes, once we estimate the radius \( r \) using the arc length formula:

• We substitute this radius back into the circumference formula.
• This gives us: \( C = 2\pi \times \frac{496 \times 180}{7.2\pi} \).

Simplifying this expression, you achieve an estimate for Earth's circumference. Understanding this relationship helps grasp how early astronomers like Eratosthenes could make such accurate estimates long before modern technology.