Convert $54.959^{\circ}$ to degree-minute-second form. If necessary, round your answer to the nearest second.
Answer (2)
The degrees part is the integer part of the angle: 5 4 ∘ .
Multiply the decimal part by 60 to get minutes: 0.959 × 60 = 57.54 , so 5 7 ′ .
Multiply the decimal part of the minutes by 60 to get seconds: 0.54 × 60 = 32.4 , which rounds to 3 2 ′′ .
Combine the results: 5 4 ∘ 5 7 ′ 3 2 ′′ .
Explanation
Understanding the Problem We are given an angle in decimal degrees, 54.95 9 ∘ , and we want to convert it to degree-minute-second (DMS) form. The DMS form breaks down an angle into degrees, minutes, and seconds, where 1 degree is equal to 60 minutes, and 1 minute is equal to 60 seconds.
Extracting Degrees The whole number part of the decimal degree is the degree part of the DMS angle. In this case, it is 5 4 ∘ .
Calculating Minutes To find the minutes, we take the decimal part of the original angle (0.959) and multiply it by 60: 0.959 × 60 = 57.54 The whole number part of this result is the minutes part of the DMS angle, which is 5 7 ′ .
Calculating Seconds To find the seconds, we take the decimal part of the minutes calculation (0.54) and multiply it by 60: 0.54 × 60 = 32.4 We are asked to round to the nearest second, so we round 32.4 to 32. Thus, the seconds part of the DMS angle is 3 2 ′′ .
Final Answer Combining the degrees, minutes, and seconds, we get the angle in DMS form: 5 4 ∘ 5 7 ′ 3 2 ′′ .
Examples
Converting angles to degree-minute-second form is useful in navigation, astronomy, and surveying. For example, when using a sextant to measure the angle to a star, the result is often in decimal degrees, but navigational charts and tables use DMS. Converting allows for easy comparison and use of the data.
To convert 54.95 9 ∘ to DMS form, extract the degrees as 5 4 ∘ , calculate minutes as 5 7 ′ by multiplying the decimal part by 60, and then calculate seconds as 3 2 ′′ from the decimal of minutes. The final result is 5 4 ∘ 5 7 ′ 3 2 ′′ .
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