To convert angles into decimal degrees, we use the following approach:
1. Convert minutes to a fraction of a degree by dividing by 60.
2. Convert seconds to a fraction of a degree by dividing by 3600 (since [tex]\(1 \text{ minute} = 60 \text{ seconds}\)[/tex]).
Let's handle each angle step by step:
### a) [tex]\(10^{\circ} 30^{\prime}\)[/tex]
1. Degrees:
- [tex]\(10^{\circ}\)[/tex]
2. Minutes:
- [tex]\(30^{\prime} = \frac{30}{60} = 0.5\)[/tex] degrees
So, adding them together:
[tex]\[ 10 + 0.5 = 10.5 \][/tex]
Thus, [tex]\(10^{\circ} 30^{\prime} = 10.5\)[/tex] degrees.
### b) [tex]\(12^{\circ} 15^{\prime} 30^{\prime \prime}\)[/tex]
1. Degrees:
- [tex]\(12^{\circ}\)[/tex]
2. Minutes:
- [tex]\(15^{\prime} = \frac{15}{60} = 0.25\)[/tex] degrees
3. Seconds:
- [tex]\(30^{\prime \prime} = \frac{30}{3600} = 0.008333333333333333\)[/tex] degrees
So, adding them together:
[tex]\[ 12 + 0.25 + 0.008333333333333333 = 12.258333333333333 \][/tex]
Thus, [tex]\(12^{\circ} 15^{\prime} 30^{\prime \prime} = 12.258333333333333\)[/tex] degrees.
### c) [tex]\(14^{\circ} 18^{\prime} 45^{\prime \prime}\)[/tex]
1. Degrees:
- [tex]\(14^{\circ}\)[/tex]
2. Minutes:
- [tex]\(18^{\prime} = \frac{18}{60} = 0.3\)[/tex] degrees
3. Seconds:
- [tex]\(45^{\prime \prime} = \frac{45}{3600} = 0.0125\)[/tex] degrees
So, adding them together:
[tex]\[ 14 + 0.3 + 0.0125 = 14.3125 \][/tex]
Thus, [tex]\(14^{\circ} 18^{\prime} 45^{\prime \prime} = 14.3125\)[/tex] degrees.
### d) [tex]\(18^{\circ} 15^{\prime} 14^{\prime \prime}\)[/tex]
1. Degrees:
- [tex]\(18^{\circ}\)[/tex]
2. Minutes:
- [tex]\(15^{\prime} = \frac{15}{60} = 0.25\)[/tex] degrees
3. Seconds:
- [tex]\(14^{\prime \prime} = \frac{14}{3600} = 0.0038888888888888888\)[/tex] degrees
So, adding them together:
[tex]\[ 18 + 0.25 + 0.0038888888888888888 = 18.253888888888888 \][/tex]
Thus, [tex]\(18^{\circ} 15^{\prime} 14^{\prime \prime} = 18.253888888888888\)[/tex] degrees.
### Summary
a) [tex]\(10^{\circ} 30^{\prime} = 10.5\)[/tex] degrees
b) [tex]\(12^{\circ} 15^{\prime} 30^{\prime \prime} = 12.258333333333333\)[/tex] degrees
c) [tex]\(14^{\circ} 18^{\prime} 45^{\prime \prime} = 14.3125\)[/tex] degrees
d) [tex]\(18^{\circ} 15^{\prime} 14^{\prime \prime} = 18.253888888888888\)[/tex] degrees
