A plane is flying 225 mph heading S25°W. The wind begins blowing S80°E at 60 mph.
What is the direction of the plane's new path using compass points such as N17°W?
What is the ground speed of the plane now?
What is the smallest angle of the triangle?
What is the largest angle of the triangle?
What is the remaining angle of the triangle?
Well, well, well, looks like we have an aerial circus act in the making! Let's unravel this flying extravaganza, shall we?
First off, to determine the direction of the plane's new path using compass points, we need to assess the effect of the wind. The plane is flying south(ish) at S25°W, but the wind decides to join the party from the opposite direction, S80°E. These two angles are like rival clowns juggling for attention!
To find the direction of the new path, we can simply add these two angles together. S25°W + S80°E is like trying to mix oil and water - it just doesn't work! So let's convert these angles into a more compatible format.
S25°W can be seen as traveling 25° to the west of south. Similarly, S80°E can be interpreted as traveling 80° to the east of south. Now, let's put these two clowns together!
Adding the two angles, we have 25° (south) + 80° (east), which gives us a new angle of E55°S. Ta-da! That's the direction of the plane's new path, according to compass points.
Now, let's move on to the ground speed of the plane. The plane was originally flying at 225 mph, but the wind pushes it around like a mischievous gust. The wind speed is 60 mph, so we need to subtract that from the plane's original speed.
225 mph - 60 mph gives us a new ground speed of 165 mph. That's how fast this circus act is zooming across the skies!
Next up, let's take a look at the triangle formed by the original path, the wind, and the new path. This triangle is like a secret language for angle enthusiasts!
The smallest angle of the triangle would be the angle between the original path and the new path. Since the original heading was S25°W and the new heading is E55°S, we can subtract these two angles.
E55°S - S25°W is like subtracting the funniest clown at the circus - it leaves us with an angle of E30°S. That's the smallest angle of the triangle!
The largest angle of the triangle would be the angle between the wind direction and the new path. With the wind blowing at S80°E and the new heading being E55°S, we can again subtract these angles.
E55°S - S80°E is like removing the grand finale clown from the circus - it leaves us with an angle of S155°W. That's the largest angle of the triangle!
Lastly, the remaining angle of the triangle would be the angle between the original path and the wind direction. We can subtract these angles to find out this mysterious angle.
S80°E - S25°W is like realizing you lost your clown nose at the circus - it leaves us with an angle of E55°E. That's the remaining angle of the triangle!
And there you have it, my friend! The plane's new path is E55°S, its ground speed is 165 mph, the smallest angle of the triangle is E30°S, the largest angle is S155°W, and the remaining angle is E55°E. What a daring performance by our aerial clown pilot!
To find the direction of the plane's new path, we need to add the vectors representing the plane's velocity and the wind's velocity.
Step 1: Decompose the wind's velocity into its components.
The wind is blowing S80°E. The angle between the south direction and the south-east direction is 90° - 80° = 10°. Therefore, the component of the wind's velocity in the south direction is 60 mph * cos(10°) = 59.81 mph, and the component in the east direction is 60 mph * sin(10°) = 10.34 mph.
Step 2: Add the components of the plane's velocity and the wind's velocity.
The plane's velocity is 225 mph heading S25°W, which means its component in the south direction is 225 mph * cos(25°) = 197.1 mph, and its component in the west direction is 225 mph * sin(25°) = 92.9 mph.
Adding the components gives us:
South component: 197.1 mph + 59.81 mph = 256.91 mph
West component: 92.9 mph - 10.34 mph = 82.56 mph
Step 3: Convert the components back into magnitude and direction.
Using the components, we can find the magnitude of the resultant velocity using the Pythagorean theorem:
Magnitude of resultant velocity = sqrt((256.91 mph)^2 + (82.56 mph)^2) ≈ 269.32 mph
To find the direction, we can use the inverse tangent function:
Direction = atan(82.56 mph / 256.91 mph) ≈ 17°
Since the west component is positive and the south component is positive, the direction is towards the fourth quadrant, which is approximately S17°W.
The ground speed of the plane is equal to the magnitude of the resultant velocity, which is approximately 269.32 mph.
To find the angles of the triangle formed by the plane's velocity, wind's velocity, and the resultant velocity, we can use the Law of Cosines.
Let A, B, and C be the angles formed by the plane's velocity, wind's velocity, and the resultant velocity, respectively. The lengths of the sides opposite to angles A, B, and C are the magnitudes of the corresponding velocities.
Using the Law of Cosines, the smallest angle (A) can be found using the formula:
cos(A) = (b^2 + c^2 - a^2) / (2bc)
cos(A) = (59.81 mph^2 + 269.32 mph^2 - 197.1 mph^2) / (2 * 59.81 mph * 269.32 mph)
A ≈ 35.92°
The largest angle (B) can be found using the same formula, considering the wind's velocity and the resultant velocity:
cos(B) = (a^2 + c^2 - b^2) / (2ac)
cos(B) = (197.1 mph^2 + 269.32 mph^2 - 59.81 mph^2) / (2 * 197.1 mph * 269.32 mph)
B ≈ 69.74°
Finally, the remaining angle (C) can be found by subtracting the sum of angles A and B from 180°:
C = 180° - A - B
C ≈ 74.34°
To summarize:
- The direction of the plane's new path is approximately S17°W.
- The ground speed of the plane is approximately 269.32 mph.
- The smallest angle of the triangle is approximately 35.92°.
- The largest angle of the triangle is approximately 69.74°.
- The remaining angle of the triangle is approximately 74.34°.
To determine the direction of the plane's new path, we first need to calculate the resultant direction. This can be done by adding the direction of the plane's initial path to the direction of the wind's effect on the plane.
1. Direction of the plane's initial path: S25°W.
2. Direction of the wind: S80°E.
To add these directions, we need to convert them to a common reference frame. Let's use compass points.
- S25°W can be converted to N65°W. (180° - 25° = 155° + 90° = 245° - 180° = 65°)
- S80°E can be converted to N10°E. (180° - 80° = 100° + 90° = 190° - 180° = 10°)
Now we can add these directions:
N65°W + N10°E = N55°W.
Therefore, the direction of the plane's new path is N55°W.
To calculate the ground speed of the plane, we need to find the magnitude of the resultant velocity vector. This can be done using the Pythagorean theorem.
1. Speed of the plane: 225 mph.
2. Speed of the wind: 60 mph.
Using the Pythagorean theorem:
Ground speed = √((speed of the plane)^2 + (speed of the wind)^2)
= √((225 mph)^2 + (60 mph)^2)
= √(50625 mph^2 + 3600 mph^2)
= √(54225 mph^2)
≈ 232.80 mph (rounded to two decimal places)
Therefore, the ground speed of the plane is approximately 232.80 mph.
To find the smallest angle of the triangle formed by the plane's initial path, the wind's effect, and the resultant path, we need to determine the angle between the plane's initial path and the direction of the wind.
Angle between the plane's initial path (N65°W) and the direction of the wind (N10°E) can be calculated as follows:
180° - |65° - 10°| = 180° - 55° = 125°
Therefore, the smallest angle of the triangle is 125°.
The largest angle of the triangle can be calculated by subtracting the smallest angle and the angle of the wind's effect (90°) from 180°.
Largest angle = 180° - 125° - 90° = 180° - 215° = -35°
Since angles cannot be negative, we take the absolute value, resulting in an angle of 35°.
Therefore, the largest angle of the triangle is 35°.
The remaining angle of the triangle can be calculated by subtracting the sum of the smallest and largest angles from 180°.
Remaining angle = 180° - 125° - 35° = 20°
Therefore, the remaining angle of the triangle is 20°.