You are taking a flight from Daytona Beach to St. Louis. There is a stopover in Atlanta. The bearing from Daytona Beach to Atlanta is N 38.3 degrees W. The bearing from Dayton directly to St. Louis is N 43.4 degrees W. The distance of the first leg of the trip from Daytona to Atlanta is approximately 599.8 Kilometers. From Atlanta to St. Louis is another 748.4 kilometers.
a. What is the total length of the trip with the stopover?
b. How much shorter would it be to fly directly to St. Louis without the stopover?
did you make a sketch?
for a) wouldn't you just add up the two distances ?
b) my diagram is ABC, with angle B = 5.1°
a = 599.8 , b=748.4
by the sine law,
sinA/599.8 = sin5.1/748.4
A = 4.01037...°
angle C = 180-5.1-4.01037... = .......
now that you have angle C, use the sine law again to find AB
take over
1343.86
To find the total length of the trip with the stopover, you need to calculate the distance from Daytona Beach to Atlanta and then add it to the distance from Atlanta to St. Louis.
a. Distance from Daytona Beach to Atlanta = 599.8 kilometers
Distance from Atlanta to St. Louis = 748.4 kilometers
Total length of the trip with stopover = Distance from Daytona Beach to Atlanta + Distance from Atlanta to St. Louis = 599.8 + 748.4 = 1348.2 kilometers
b. To find out how much shorter the direct flight to St. Louis would be without the stopover, you need to subtract the distance from Daytona Beach to Atlanta from the total length of the trip with the stopover.
Shorter distance without the stopover = Total length of the trip with stopover - Distance from Daytona Beach to Atlanta = 1348.2 - 599.8 = 748.4 kilometers
To determine the total length of the trip with the stopover and the shorter distance without the stopover, we can use trigonometry and basic arithmetic calculations.
a. To find the total length of the trip with the stopover, we can treat the distances from Daytona Beach to Atlanta and Atlanta to St. Louis as two sides of a triangle. We can use the Law of Cosines to find the length of the third side, which represents the total length of the trip.
The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case:
a = 599.8 kilometers (Daytona Beach to Atlanta)
b = 748.4 kilometers (Atlanta to St. Louis)
C = 180 - (bearing from Daytona to Atlanta - bearing from Daytona to St. Louis)
= 180 - (38.3 - 43.4) degrees
Calculate the value of C in degrees, then convert it to radians:
C_rad = C_deg * (pi / 180)
Next, substitute these values into the formula to find the total length of the trip (c).
c^2 = (599.8)^2 + (748.4)^2 - 2 * 599.8 * 748.4 * cos(C_rad)
By taking the square root of both sides, we can find the total length of the trip.
c = sqrt((599.8)^2 + (748.4)^2 - 2 * 599.8 * 748.4 * cos(C_rad))
b. To find the shorter distance without the stopover, we can simply use the distance from Atlanta to St. Louis.
Therefore, the distance without the stopover is 748.4 kilometers.
You can now calculate the values of a, b, C, and C_rad, and use these values to find the total length of the trip with the stopover (a), and the shorter distance without the stopover (b) using the formulas provided.