Problem 30 - Algebra 2

Question

Geography. The 24 lines of longitude that approximate the 24 standard time zones are equally spaced around the equator. a. Suppose you use 24 central angles to divide a circle into 24 equal arcs. Express the measure of each angle in degrees and in radians. b. The radius of the equator is about 396 mi. About how wide is each time zone at the equator? c. The radius of the Arctic Circle is about 1580 mi. About how wide is each time zone at the Arctic Circle?

Step-by-Step Solution

Answer

The measure of each central angle used to create 24 equal arcs around a circle is 15 degrees or \( \frac{\pi}{12} \) radians. Each time zone at the equator and the Arctic Circle is approximately \( 3960 \times \frac{\pi}{12} \) miles and \( 1580 \times \frac{\pi}{12} \) miles wide, respectively.

Step 1: Determine Central Angle

Each standard time zone corresponds with a line of longitude, and there are 24 of these around the whole equator, which is a circle. Dividing a full circle (360 degrees) by 24, obtain the measure of each central angle in degrees, which is \( \frac{360}{24} = 15 \) degrees.

Step 2: Convert to Radians

To convert 15 degrees to radians, use the formula \( radians = degrees \times \frac{\pi}{180} \). Thus, each angle measures \( 15 \times \frac{\pi}{180} = \frac{\pi}{12} \) radians.

Step 3: Calculate Width of Each Time Zone at the Equator

The width of each time zone at the equator is equivalent to the arc length of the central angle on the circle representing the equator. The formula to find the arc length (s) is \( s = rθ \) where r = radius of the equator (~3960 miles), and θ = angle in radians. So, \( s = 3960 \times \frac{\pi}{12} \) miles.

Step 4: Calculate Width of Each Time Zone at the Arctic Circle

Repeating the process in Step 3, but this time using the radius of the Arctic Circle (~1580 miles), the width of each time zone at the Arctic Circle equals \( s = 1580 \times \frac{\pi}{12} \) miles.

Key Concepts

Central Angles

Central angles are essential in understanding how we divide a circle into different sections, particularly when dealing with time zones. A central angle is formed by two rays originating from the center of a circle, intersecting the circumference. For a full circle, the total central angle measures 360 degrees.
In the context of time zones, the Earth is divided into 24 such angles, each corresponding to a time zone. By dividing 360 degrees by 24, we find that each central angle equates to 15 degrees.

• This division represents how the Earth is segmented into different time zones for standardized time-keeping.
• Each segment not only represents a slice of the circle but also a span of one hour in time.

Understanding central angles helps us grasp the intricate relationship between geography and time measurement across the globe.

Arc Length

Arc length is the distance between two points along a section of a curve or circle. In the study of time zones, arc length can determine the width of each zone at various points on Earth. This is particularly helpful when examining the equator and the Arctic Circle.

The formula for arc length is given by \[ s = r \theta \]where \(s\) represents the arc length, \(r\) is the radius of the circle, and \(\theta\) is the central angle in radians.

• At the equator, where the radius is roughly 3960 miles, the arc length of each time zone is the product of 3960 and \(\frac{\pi}{12}\).
• At the Arctic Circle, the radius decreases to approximately 1580 miles. Thus, the arc length for each time zone here is 1580 multiplied by \(\frac{\pi}{12}\).

By calculating these arc lengths, we can visualize the physical width of time zones at different latitudes and understand how geography affects time zone boundaries.

Radian Conversion

Radians provide an alternative way to measure angles, more naturally aligning with mathematical computations involving circles. To convert degrees into radians, we use the formula: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180}\]This conversion is crucial when working with formulas related to arcs and areas of circles.

For time zones, each central angle of 15 degrees is equivalent to \(15 \times \frac{\pi}{180} = \frac{\pi}{12}\) radians.

• Radians simplify the computation when plugged into formulas for finding arc lengths or when calculating areas.
• It allows direct applicability in sectors of calculus and trigonometry where circular measurements are common.

Understanding radian conversion enhances the way we solve problems involving circular dimensions, offering a more seamless approach to mathematical operations related to angles.

Longitude and Latitude

Longitude and latitude coordinate the Earth into a grid system, essential for navigation and time zone calculations. Longitudes are imaginary lines running from pole to pole, while latitudes run parallel to the equator.

• Longitude lines help define time zones by dividing the Earth into 24 sections of 15 degrees each, representing different time zones.
• Latitude lines are key when calculating the arc length of time zones at different points, like the equator and Arctic Circle, due to varying curvature.

By understanding longitude and latitude, one can grasp how geographic locations contribute to the division of time zones. These concepts also play a pivotal role in fields like aviation and maritime navigation, impacting global communication and transportation.