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Chapter 4: Problem 30

Change the given angles to equal angles expressed to the nearest minute. $$142.87^{\circ}$$

Short Answer

Expert verified
The angle is \(142^{\circ}52'\) when expressed to the nearest minute.

Step by step solution

01

Understand the Problem

We need to convert the given angle in decimal degrees to degrees and minutes format, expressing the answer to the nearest minute. The angle provided is \(142.87^{\circ}\).
02

Separate Degrees from Minutes

First, we identify the degree part, which is \(142\). The remaining part after removing the integers is \(0.87\). This fractional part represents a portion of a degree that needs to be converted to minutes.
03

Convert Decimal Degrees to Minutes

Multiply the decimal portion by 60 to convert it into minutes. Calculate \(0.87 \times 60 = 52.2\) minutes.
04

Round to the Nearest Minute

Since we need to round \(52.2\) to the nearest minute, we look at the tenths digit. If it is 5 or greater, we round up. Here, \(52.2\) rounds to \(52\) minutes.
05

Combine Degrees and Minutes

Now combine the integer degrees with the rounded minutes: \(142^{\circ}\) and \(52'\). So, the angle expressed to the nearest minute is \(142^{\circ}52'\).

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Key Concepts

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

Understanding Decimal Degrees
Decimal degrees represent an angle using a single number, which is a more modern and straightforward way to express angles compared to degrees, minutes, and seconds (DMS). This format makes it easier for calculations and can be directly used in many mathematical and engineering applications.
Decimal degrees take the full angle where:
  • The whole number part represents the degrees.
  • The decimal part signifies the fraction of a degree.
For example, an angle of 142.87 degrees consists of 142 full degrees, and the 0.87 represents the fractional part of the next degree. As calculations with decimal degrees are straightforward in many software tools, they are heavily used in GPS and mapping technologies.
Degrees to Minutes Conversion
Degrees can be broken down into smaller units for greater precision, with minutes being one of them. One degree is equivalent to 60 minutes (\[1^{\circ} = 60'\]). This division allows us to express parts of a degree that are not whole in a more precise manner.
To convert from decimal degrees to a degree and minutes format, you first identify the decimal fraction, such as 0.87 from 142.87. You then multiply this by 60 (since there are 60 minutes in a degree) to get the number of minutes. For our example:
  • Find the fractional part: 0.87
  • Convert to minutes: \[0.87 \times 60 = 52.2'\]
This tells you that 0.87 of a degree is equivalent to approximately 52.2 minutes.
Rounding Angles
Rounding is crucial when dealing with fractional parts of angles to make them easier to express and understand. When converting decimal degrees to minutes, you often need to round to the nearest whole minute.
The rule for rounding is straightforward:
  • If the first digit after the decimal is 5 or more, you round the number up.
  • If it is less than 5, you round the number down.
Thus, when you have 52.2 minutes from the conversion of 142.87 degrees, you check the decimal (0.2). Since it's less than 5, you round the 52.2 down to 52 minutes. This ensures that the angle, now expressed as 142 degrees and 52 minutes, is both accurate and neatly expressed for practical use.

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