Problem 66 - Precalculus

Question

Find the distance along an an an the surface of Earth that subtends a central angle of 7 minutes \(\left(1 \text { minute }=\frac{1}{60} \text { degree). The radius of Earth is } 3960 \text { miles. }\right.\)

Step-by-Step Solution

Answer

The distance is approximately 8.07 miles.

Step 1: Convert Minutes to Degrees

The central angle is given in minutes, so we need to convert this value into degrees. Since 1 degree equals 60 minutes, we can convert 7 minutes into degrees using the formula:\[\text{Degrees} = \frac{7}{60} = \frac{7}{60} \text{ degrees}.\]

Step 2: Calculate the Arc Length

The arc length of a circle is given by the formula: \( \text{Arc Length} = \text{Radius} \times \text{Central Angle in Radians} \). First, convert the central angle from degrees to radians by using the formula:\[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]Substitute the degree from Step 1:\[\text{Radians} = \frac{7}{60} \times \frac{\pi}{180} = \frac{7\pi}{10800}\]Now, calculate the arc length:\[\text{Arc Length} = 3960 \times \frac{7\pi}{10800}\]Simplify the expression:\[\text{Arc Length} \approx 3960 \times 0.002037 = 8.07392 \text{ miles}\]

Step 3: Provide the Final Answer

Rounding the answer for practical purposes, the distance along the Earth's surface that subtends a central angle of 7 minutes is approximately 8.07 miles.

Key Concepts

Central Angle

A central angle is key in circle geometry. It's the angle formed by two radii extending from the center of the circle to the edge. The unique aspect of central angles is that they directly define the size of the arc that they intersect on the circle's circumference.
A central angle is always measured at the center of the circle, defining a specific section of the circle.
In the context of the Earth's surface, which is treated as a circular arc due to its round shape, the central angle helps determine how far we need to travel along the surface for a particular section. This is the heart of our exercise: finding the distance along the Earth's surface given a central angle of 7 minutes, converted via the rules of degree to radian conversion.

Degree to Radian Conversion

Angles can be expressed in many units, but degrees and radians are the most common. In mathematics and science, radians are often preferred. This preference exists because radian measures are based on the properties of circles, allowing simple calculations like arc length.
To switch between the two, use the conversion formula:

• Degrees to Radians: \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \)
• Radians to Degrees: \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \)

In our exercise, the central angle was initially in minutes. Since 1 degree equals 60 minutes, we first converted minutes to degrees, then to radians using the \(\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\) formula. Thus, calculations on the circle such as arc lengths become straightforward once the angle is in radians.

Radius of Earth

The radius is a measure from a circle's center to any point on its edge, and for Earth, it's quite substantial. In this exercise, we're given the radius of Earth as 3960 miles. This constant allows us to assess distances across Earth, especially useful for geographical calculations.
Knowing Earth's radius enables us to apply formulas for circles to real-world problems, like determining the span of an arc on Earth's surface. This exercise illustrates how using the Earth's radius enables the transition of mathematical principles to practical applications, such as estimating travel distances along its surface.

Circle Geometry

Circle geometry is the basis for our understanding of circular shapes and motions. It involves the properties and relations of points on a circle, including concepts like radius, diameter, circumference, and central angles.
One critical aspect of circle geometry is calculating arc length, especially useful in real-world scenarios like navigation or measuring distances. The foundational formula is:

• Arc Length = Radius × Central Angle (in radians)

The circle's geometry connects with conversion between degrees and radians and introduces practical challenges, such as approximating distances on a spherical surface like Earth. This mix of theoretical knowledge and practical application enables students to see how circle geometry functions beyond pure mathematics.