An airplane flies 200 km due west from city A to city B and then 280 km in the direction of 30.5° north of west from city B to city C.
(a) In straight-line distance, how far is city C from city A?
(b) Relative to city A, in what direction is city C?
The answer is: 17.8 degrees north of west
I received help with this question before, but I keep getting the wrong answer for (a).
Show us what you did.
okay: I got the vector from A to B and then the vector from B to C to get the distance from A to C.
I got 200.00 from the vector from A to B and then 280.00 from the vector from B to C. I added them, but the answer is incorrect.
For A to B: I did 200cos30.5
200sin30.5
Then I did the pythagorean and got 200.00
For B to C: I did 280cos17.8
280sin17.8
Pythagorean and got 280.
To find the straight-line distance between City A and City C, we can use the Pythagorean theorem. Let's break down the given information:
- Distance from City A to City B (west): 200 km
- Distance from City B to City C (northwest): 280 km
- Angle of 30.5° north of west
(a) Using the Pythagorean theorem, we can find the straight-line distance between City A and City C:
Distance^2 = (Distance from A to B)^2 + (Distance from B to C)^2
Distance^2 = 200^2 + 280^2
Distance^2 = 40,000 + 78,400
Distance^2 = 118,400
Distance ≈ √(118,400)
Distance ≈ 344.13 km
Therefore, the straight-line distance between City A and City C is approximately 344.13 km.
(b) To determine the direction of City C relative to City A, we can use trigonometric functions. Since the angle is given as north of west, we subtract it from 90° to get the angle relative to west:
Angle = 90° - 30.5°
Angle = 59.5°
Therefore, City C is approximately 59.5° north of west relative to City A.
To find the straight-line distance between City A and City C, you can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the airplane flies 200 km due west from City A to City B, and then 280 km in the direction of 30.5° north of west from City B to City C.
First, let's find the horizontal distance traveled from City A to City B (which is due west). This horizontal distance is equal to the 200 km traveled in the x-direction.
Next, let's find the vertical distance traveled from City B to City C (which is in the direction of 30.5° north of west). This vertical distance is equal to the 280 km traveled in the y-direction.
To find the straight-line distance, we need to find the resultant distance (the hypotenuse) by using the horizontal and vertical distances.
(a) Straight-line distance between City A and City C:
Using the Pythagorean theorem, we can write:
(horizontal distance)^2 + (vertical distance)^2 = (straight-line distance)^2
Let's calculate the horizontal distance:
(horizontal distance)^2 = (200 km)^2 = 40000 km^2
Now, let's calculate the vertical distance:
(vertical distance) = (280 km * sin(30.5°)) = 147.33 km
(vertical distance)^2 = (147.33 km)^2 = 21679.11 km^2
Now, let's find the square root of the sum of the squares of the horizontal and vertical distances to find the straight-line distance:
(straight-line distance)^2 = (40000 km^2) + (21679.11 km^2) = 61679.11 km^2
(straight-line distance) = sqrt(61679.11 km^2) = 248.15 km
So, the straight-line distance between City A and City C is approximately 248.15 km.
(b) Relative to City A, in what direction is City C:
To find the direction, we can use trigonometry. The direction can be represented as an angle measured counterclockwise from the west direction.
The horizontal distance traveled from City A to City B is 200 km, and the vertical distance traveled from City B to City C is 280 km * sin(30.5°) = 147.33 km.
To find the angle, we use the tangent function:
Angle = atan(vertical distance ÷ horizontal distance)
Angle = atan(147.33 km ÷ 200 km)
Angle ≈ 35.84°
However, since the angle is measured counterclockwise from the west, we need to subtract this angle from 180° to get the angle relative to City A:
Relative angle = 180° - 35.84° ≈ 144.16°
The city C is located in the northwest direction of City A since the angle is measured from the west direction. To specify the direction more precisely, we need to subtract 90° from the angle:
Relative angle = 144.16° - 90° ≈ 54.16°
So, relative to City A, City C is approximately 54.16° north of west.
Therefore, the answer is:
(a) The straight-line distance between City A and City C is approximately 248.15 km.
(b) Relative to City A, City C is approximately 54.16° north of west.