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Chapter 8: Problem 35

Convert to decimal degree notation. Round to the nearest hundredth. $$18^{\circ} 14^{\prime} 20^{\prime \prime}$$

Short Answer

Expert verified
18.24掳

Step by step solution

01

- Understand the given notation

The given angle is in degrees, minutes, and seconds: \(18^{\bullet} 14^{\backprime} 20^{\backprime \backprime}\). Degrees (掳), minutes (鈥�), and seconds (鈥�) are units to measure angles, where: 1掳 = 60鈥� 1鈥� = 60鈥�.
02

- Convert seconds to decimal degrees

Divide the seconds by the number of seconds in one degree: \(\frac{20}{3600}\). This gives: \(20^{\backprime \backprime} \times \frac{1^{\backprime}}{60^{\backprime \backprime}} \times \frac{1^{\bullet}}{60^{\backprime}} = \frac{20}{3600} \ = 0.00556^{\bullet} \).
03

- Convert minutes to decimal degrees

Divide the minutes by the number of minutes in one degree: \(\frac{14}{60}\). This gives: \(14^{\backprime} \times \frac{1^{\bullet}}{60^{\backprime}} = \frac{14}{60} = 0.23333^{\bullet}\).
04

- Add all the contributions

Add the degrees, the decimal degrees obtained from the minutes, and the decimal degrees obtained from the seconds: \[18^{\bullet} + 0.23333^{\bullet} + 0.00556^{\bullet} = 18.23889^{\bullet}\]
05

- Round the result

Round the result to the nearest hundredth:\[18.23889^{\bullet} \rightarrow 18.24^{\bullet}\]

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Key Concepts

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

Angle Measures
Angles are a fundamental concept in geometry and trigonometry. They are used to describe the amount of turn or rotation between two lines or planes that meet at a common point, called the vertex.
The three primary units used to measure angles are:
  • Degrees (掳)
  • Minutes (鈥�)
  • Seconds (鈥�)
Degrees are the most common unit and are represented by the symbol 掳. A full circle is divided into 360 degrees. Sometimes, angles are given in minutes and seconds for more precision. There are 60 minutes in one degree and 60 seconds in one minute. Understanding how to measure and convert these angles is crucial in various fields, such as navigation, astronomy, and engineering.
Degrees, Minutes, and Seconds
When dealing with angle measures, it's essential to understand the notation of degrees, minutes, and seconds.
For example, an angle might be given as: $$18^{\bullet} 14^{\backprime} 20^{\backprime \backprime}$$.
This represents:
  • 18 degrees
  • 14 minutes
  • 20 seconds
To convert this to a single decimal degree, you need to convert both minutes and seconds into fractions of a degree and then add them together.
The conversion rates are:
  • 1 minute (鈥�) = \frac{1}{60} degrees
  • 1 second (鈥�) = \frac{1}{3600} degrees
Using these conversion rates makes it easy to change an angle from degrees, minutes, and seconds into decimal degrees.
Decimal Notation
Decimal notation is a way of expressing numbers that includes a decimal point to indicate a fraction of a whole. When converting angle measures from degrees, minutes, and seconds to decimal degrees, the goal is to express all parts of the angle as a single continuous number.
Here's how to convert the example angle $$18^{\bullet} 14^{\backprime} 20^{\backprime \backprime}$$ to decimal degrees:
  • Convert the seconds (20鈥�) to degrees: \frac{20}{3600} \rightarrow 0.00556^{\bullet}
  • Convert the minutes (14鈥�) to degrees: \frac{14}{60} \rightarrow 0.23333^{\bullet}
  • Add these to the degrees (18掳): 18 + 0.23333 + 0.00556 = 18.23889
Finally, round this number to the nearest hundredth to get the final answer: 18.24掳. This method can simplify comparing and calculating with angles, making it a useful skill in many math and science applications.

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