Questions: Question 6 Convert the angle 309.18666666667° to degrees/minutes/seconds form. Question 7 Subtract the two angles in degrees/minutes/seconds form. 70° 49′ 47″ - 246° 59′ 36″ Question 8

Solution Steps

For Question 6:

1. Separate the integer part (degrees) from the decimal part.
2. Convert the decimal part to minutes by multiplying by 60.
3. Separate the integer part of the minutes and convert the remaining decimal part to seconds by multiplying by 60.

For Question 7:

1. Convert both angles to seconds.
2. Subtract the second angle's total seconds from the first angle's total seconds.
3. Convert the result back to degrees, minutes, and seconds.

To convert the angle \( 309.18666666667^{\circ} \):

• The integer part (degrees) is \( 309 \).
• The decimal part is \( 0.18666666667 \).
• Convert the decimal part to minutes: \[ 0.18666666667 \times 60 \approx 11.2000000002 \implies 11 \text{ minutes} \]
• Convert the remaining decimal part to seconds: \[ 0.2000000002 \times 60 \approx 12.0000000121 \implies 12.00 \text{ seconds} \] Thus, \( 309.18666666667^{\circ} \) is equivalent to \( 309^{\circ} 11' 12.00'' \).

To subtract \( 70^{\circ} 49' 47'' \) from \( 246^{\circ} 59' 36'' \):

• Convert both angles to total seconds: \[ \text{Total seconds for } 70^{\circ} 49' 47'' = 70 \times 3600 + 49 \times 60 + 47 = 254987 \text{ seconds} \] \[ \text{Total seconds for } 246^{\circ} 59' 36'' = 246 \times 3600 + 59 \times 60 + 36 = 889176 \text{ seconds} \]
• Subtract the second angle from the first: \[ 254987 - 889176 = -634189 \text{ seconds} \]
• Convert the result back to degrees, minutes, and seconds: \[ \text{Degrees} = \left\lfloor \frac{-634189}{3600} \right\rfloor = -177 \] \[ \text{Remaining seconds} = -634189 \mod 3600 = 3011 \] \[ \text{Minutes} = \left\lfloor \frac{3011}{60} \right\rfloor = 50 \] \[ \text{Seconds} = 3011 \mod 60 = 11 \] Thus, the result of the subtraction is \( -177^{\circ} 50' 11'' \).

Final Answer