Answer :
Answer: 870.11 miles ( approx )
Step-by-step explanation:
The difference between the latitudes of the cities = 41° 15' 50'' N - 32° 47' 39''
= [tex](41^{\circ}+(\frac{15}{60})^{\circ}+(\frac{50}{3600})^{\circ})-(32^{\circ}+(\frac{47}{60})^{\circ}+(\frac{39}{3600})^{\circ})[/tex]
= [tex]9^{\circ}-(\frac{32}{60})^{\circ}+(\frac{11}{3600})^{\circ}[/tex]
= [tex](\frac{44891}{3600})^{\circ}[/tex]
= [tex](\frac{44891}{3600})\times \frac{\pi}{180^{\circ}}\text{ radian}[/tex] [tex]( 1 \pi = 180^{\circ} )[/tex]
= [tex]\frac{44891\pi}{648000}\text{ Radian}[/tex]
Also, the radius of the earth = 4000 miles
Hence, the distance between the cities
[tex] = \frac{44891\pi}{648000}\times 4000[/tex]
[tex]=\frac{179564000\pi}{648000}[/tex]
[tex]=277.104938\pi[/tex]
[tex]=277.104938\times 3.14=870.109505\approx 870.11\text{ miles}[/tex]
The angular difference between two cities is converted to its linear
equivalent at the given radius to find the distance between the cities.
The distance between the cities is approximately 591.3 miles.
Reasons:
The latitude of the given cities are;
Dallas, Texas; 32° 47' 39'' N
Omaha, Nebraska; 41° 15' 50'' N
The location of the longitude of the given latitude = The same
Required:
The distance between the given cities
Solution:
The difference between the given latitudes is found as follows;
[tex]{}[/tex] 41° 15' 50''
- 32° 47' 39''
[tex]{}[/tex] 8° 28' 11''
Where;
1 minute, 1' = [tex]\dfrac{1^{\circ}}{60}[/tex]
1 second, 1'' = [tex]\dfrac{1^'}{60}[/tex]
Converting to degrees gives;
[tex]\left(8 + \dfrac{28}{60} +\dfrac{11}{3600} \right) ^{\circ} = 8.4697 \overline 2 ^{\circ}[/tex]
Converting to radians gives;
π radians = 180°
[tex]1^{\circ} = \dfrac{\pi}{180}[/tex]
[tex]\left(8 + \dfrac{28}{60} +\dfrac{11}{3600} \right) ^{\circ} = \dfrac{\pi}{180} \times \left(8 + \dfrac{28}{60} +\dfrac{11}{3600} \right) \ radians[/tex]
Linear distance = Radius × Angular distance
The distance between the two cities is therefore;
[tex]4000 \ miles \times \left(\dfrac{\pi}{180} \times \left(8 + \dfrac{28}{60} +\dfrac{11}{3600} \right)\right) \ radians = \dfrac{30491\cdot \pi}{162} \ miles \approx 591.3 \ miles[/tex]
The distance between the cities is approximately 591.3 miles.
Learn more here:
https://brainly.com/question/16728619