An airplane is leaving the first airport (A) at a bearing of 70 degrees. It then travels 238 miles to airport B. After fueling up, it leaves at a bearing of 120 degrees and lands at airport C. After fueling up it finally returns to airport (A) at a bearing of 260 degrees. What is the total miles of the complete trip?
Put your answer to 1 decimal place into answerblank 1 (no spaces)
change "bearing" to "heading" throughout.
Draw a diagram. Angle CAB = 60°, ABC = 90° and BCA = 30°
so, you have a 30-60-90 triangle, with the long leg=238.
short leg = 238/√3 = 137
and hypotenuse twice that or 274
so, the whole trip is 238+137+274=649 miles
To find the total miles of the complete trip, we need to calculate the distances traveled between each airport.
1. Distance from A to B:
The airplane traveled 238 miles from airport A to airport B.
2. Distance from B to C:
Since the bearing of departure from airport B is 120 degrees and the bearing of arrival at airport C is not given, we cannot calculate the distance directly. However, we can use the Law of Cosines to find the distance.
Let's assume that the angle between the bearings of departure and arrival (from B to C) is X (in degrees). Then we can use the following formula:
c^2 = a^2 + b^2 - 2ab * cos(X)
Where:
c = distance from B to C (unknown)
a = distance from A to B (238 miles)
b = distance from C to A (unknown)
Since the airplane returns to airport A from airport C, the sum of the distances from B to C and from C to A is 238 miles:
c + b = 238
Let's solve these two equations simultaneously:
238^2 = 238^2 + b^2 - 2(238)(b) * cos(X)
Cross-multiplying and simplifying:
0 = b^2 - 2(238)(b) * cos(X)
b(b - 2(238) * cos(X)) = 0
Since the distance cannot be negative, b = 2(238) * cos(X)
Substituting this value back into the equation c + b = 238:
c + 2(238) * cos(X) = 238
c = 238 - 2(238) * cos(X)
c = 238(1 - 2cos(X))
Now, to calculate the total distance, we need to find the value of cos(X).
3. Distance from C to A:
The bearing from C to A is 260 degrees. We can use the same approach as we did for calculating the distance from B to C.
Let's assume that the angle between the bearings of departure and arrival (from C to A) is Y (in degrees). Then we can use the Law of Cosines:
a^2 = b^2 + c^2 - 2bc * cos(Y)
Where:
a = distance from C to A (unknown)
b = distance from B to C (238 miles)
c = distance from A to B (unknown)
Since the airplane returns to airport A from airport C, the sum of the distances from B to C and from C to A is 238 miles:
a + b = 238
Let's solve these two equations simultaneously:
(238)^2 = (238)^2 + a^2 - 2(238)(a) * cos(Y)
Cross-multiplying and simplifying:
0 = a^2 - 2(238)(a) * cos(Y)
a(a - 2(238) * cos(Y)) = 0
Since the distance cannot be negative, a = 2(238) * cos(Y)
Substituting this value back into the equation a + b = 238:
2(238) * cos(Y) + b = 238
2(238) * cos(Y) = 238 - 238
2(238) * cos(Y) = 0
Since cos(Y) cannot be zero, this equation has no solution.
Therefore, the total miles of the complete trip is the distance from A to B, which is 238 miles.
Answer: 238
To solve this question, we can break down the trip into three different legs and calculate the distance for each leg.
First, let's calculate the distance for the leg from airport A to airport B. We know that the airplane travels 238 miles in this leg. So the distance for this leg is 238 miles.
Next, let's calculate the distance for the leg from airport B to airport C. We are given the bearing of 120 degrees for this leg, but we don't know the distance. To calculate the distance, we need to know the angle between the two bearings and the distance traveled in the first leg.
To find the angle between the bearings, we can subtract the initial bearing (70 degrees) from the final bearing (120 degrees). The angle between the bearings is 120 - 70 = 50 degrees.
Now, we can use the Law of Cosines to find the distance for this leg. The Law of Cosines states that c^2 = a^2 + b^2 - 2ab*cos(C), where c is the unknown side (distance for this leg), a is the distance for the first leg (238 miles), and b is the distance for the third leg (unknown). C is the angle between the two bearings (50 degrees).
Plugging in the values, we get:
c^2 = (238)^2 + b^2 - 2*238*b*cos(50)
Simplifying the equation, we have:
c^2 = 238^2 + b^2 - 2*238*b*cos(50)
c^2 = 56644 + b^2 - 2*238*b*(-0.64279)
c^2 = 56644 + b^2 + 305.62b
Since we don't know the value of b, we cannot directly solve for c. Therefore, we will move on to the third leg and come back to solve for c later.
Now, let's calculate the distance for the leg from airport C back to airport A. We are given the bearing of 260 degrees for this leg, but we don't know the distance. To calculate the distance, we need to know the angle between the two bearings, the distance traveled in the first leg, and the distance traveled in the second leg.
To find the angle between the bearings, we can subtract the second bearing (120 degrees) from the final bearing (260 degrees). The angle between the bearings is 260 - 120 = 140 degrees.
Now, we can use the Law of Cosines again to find the distance for this leg. The Law of Cosines states that c^2 = a^2 + b^2 - 2ab*cos(C), where c is the unknown side (distance for this leg), a is the distance for the second leg (unknown), and b is the distance for the third leg (unknown). C is the angle between the two bearings (140 degrees).
Plugging in the values, we get:
c^2 = a^2 + b^2 - 2ab*cos(140)
Since we don't know the value of a or b for the second leg and the value of a, b, or c for the third leg, we cannot directly solve for c.
Now, let's go back and solve for the distance of the second leg. We had the equation:
c^2 = 56644 + b^2 + 305.62b
To find the value for b, we need to solve this quadratic equation. Rearranging the equation, we get:
b^2 + 305.62b + 56644 - c^2 = 0
We can use the quadratic formula to solve for b:
b = (-305.62 ± √[(305.62)^2 - 4*1*(56644 - c^2)]) / 2*1
We have two possible solutions for b, one with the positive square root and one with the negative square root. However, since the distance cannot be negative in this context, we can discard the negative solution. Therefore, we have:
b = (-305.62 + √[(305.62)^2 - 4*1*(56644 - c^2)]) / 2*1
Now, we can substitute this value of b into the equation for the first leg:
c^2 = 56644 + b^2 + 305.62b
Substituting the value of b, we have:
c^2 = 56644 + ((-305.62 + √[(305.62)^2 - 4*1*(56644 - c^2)]) / 2)^2 + 305.62*((-305.62 + √[(305.62)^2 - 4*1*(56644 - c^2)]) / 2)
Simplifying the equation, we have a quadratic equation in terms of c. Solving this equation will give us the distance for the second leg, which we can then add up with the distances for the first and third legs to get the total miles of the trip.
You can solve this equation using numerical methods or a calculator with a quadratic solver to find the value of c. Once you have the value of c, you can add up the distances for all three legs to get the total miles of the trip.