Problem 27 - Precalculus: Functions and Graphs

Question

Exer. \(25-28:\) Express the angle in terms of degrees, minutes, and seconds, to the nearest second. $$310.6215^{\circ}$$

Step-by-Step Solution

Answer

The angle is 310° 37' 17".

Step 1: Understand the Components

The angle given is 310.6215 degrees. It needs to be expressed in degrees, minutes, and seconds. Remember, 1 degree is equal to 60 minutes, and 1 minute is equal to 60 seconds.

Step 2: Extract Whole Degrees

The whole number part, 310, is the degree component of the angle as it stands in full degrees. Therefore, we can state that the degree portion of the angle is \[ 310^{\circ} \]

Step 3: Calculate Minutes

Subtract 310 from 310.6215 to find the decimal portion of the degrees: \[ 310.6215 - 310 = 0.6215 \]Now, convert this decimal portion into minutes by multiplying by 60. \[ 0.6215 \times 60 = 37.29 \]Thus, the minute component is 37 full minutes.

Step 4: Calculate Seconds

To find the seconds, take the decimal part remaining from the minutes, which is 0.29, and convert it into seconds by multiplying by 60: \[ 0.29 \times 60 = 17.4 \]Round this to the nearest whole number, giving 17 seconds.

Step 5: Combine All Parts

Combine the degrees, minutes, and seconds to express the angle in DMS (Degrees, Minutes, Seconds): \[ 310^{\circ} 37' 17'' \]

Key Concepts

Degrees to Minutes Conversion

When converting degrees to minutes, keep in mind that the concept is based on subdivision. Each degree is equivalent to 60 minutes. This creates a seamless transition from the larger unit of measure (degrees) to the finer unit (minutes).
Here’s how it works:

• Identify the decimal portion of your degree value. For example, in 310.6215, the decimal is 0.6215.
• Multiply this decimal by 60 to convert it into minutes. For instance,
  \(0.6215 \times 60 = 37.29\).

The result is the number of minutes, with any further decimal representing a fraction of a minute that will be converted into seconds in the next step. Thus, from our example, you get 37 minutes.

Minutes to Seconds Conversion

Moving from minutes to seconds, you'll apply the same kind of subdivision. Just like a degree holds 60 minutes, a minute holds 60 seconds.
Once you have the minutes in your conversion process:

• Take the decimal fraction remaining after converting to minutes. Using our example from before, the decimal from 37.29 is 0.29.
• Multiply this decimal by 60 to find the seconds:
  \(0.29 \times 60 = 17.4\) seconds.

The resulting number, 17.4, tells you how many seconds past the initial 37 minutes your original angle extends. The decimal portion (.4 in this case) will be your focus when rounding to the nearest second.

Decimal to DMS Conversion

Converting from a decimal representation of an angle to Degrees, Minutes, and Seconds (DMS) brings together all the previous steps.
Starting with the full angle:

• The whole number portion represents the degree value. In 310.6215, that is 310°.
• The decimal is then multiplied by 60, giving the minutes. Here, 0.6215 degrees turns into 37.29 minutes.
• The decimal part of the minutes, 0.29, gets converted to seconds via another multiplication by 60, resulting in 17.4 seconds.

Now you have the angle expressed in its three-part DMS format: \(310^{\circ} 37' 17''\). It effectively organizes the angle into degrees, minutes, and seconds, making it easier to utilize in navigation and precise measurements.

Rounding to Nearest Second

Rounding is a crucial skill in converting angles because it increases ease and reduces complexity by approximating seconds to whole numbers.
Here's how you round to the nearest second:

• Look at the decimal component of your seconds. From before, we got 17.4 seconds.
• Decide based on usual rounding rules: if the decimal is 0.5 or higher, round up; if less, round down.

In our case, 17.4 becomes simply 17 seconds when rounded to the nearest whole number. Rounding is particularly useful in making final calculations cleaner and easier to share and interpret.