It is 4.7km from Lighthouse A to Port B. The bearing of the port from the lighthouse is N73E. A ship has sailed due west from the port and its bearing from the lighthouse is N31E. How far has the ship sailed from the port?
If the ship has sailed to point C,
angle A = 73°-31° = 42°
angle B = 17°
so, angle C is 121°
the distance AC is side a, where
a/sin42° = 4.7/sin121°
a = 3.67
Well, it seems like our ship took quite an interesting turn! To calculate how far the ship has sailed from the port, we need to take an amusing detour through some trigonometry.
First, let's find the angle between the bearing of the port from the lighthouse (N73E) and the bearing of the ship from the lighthouse (N31E). We can do this by subtracting the two angles: 73 - 31 = 42 degrees.
Now, we know that the ship sailed due west from the port, so we have a right-angled triangle formed by the lighthouse, the ship's position, and the port. The angle between the two bearings (42 degrees) represents the angle at the ship.
To find the distance the ship has sailed from the port, we can use a little trigonometric magic called the cosine rule. But before we do that, let's grab our popcorn and delight in the fact that math can be so twistedly entertaining!
The cosine rule states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus two times the product of the lengths of those two sides, multiplied by the cosine of the angle between them. Phew, that's a mouthful!
So, let's plug in the numbers! The side opposite to the angle we found (42 degrees) is the distance we're looking for, let's call it 'x.' The side adjacent to the angle (42 degrees) is the distance from the lighthouse to the port (4.7 km). The hypotenuse of our triangle (the distance between the lighthouse and the ship) is also 'x.'
Applying the cosine rule, we have:
x² = (4.7)² + (x)² - 2(4.7)(x)cos(42)
Solving this math riddle, we find that x ≈ 6.093 km. So, our ship has sailed approximately 6.093 km from the port.
Now, wasn't that journey through trigonometry hilarious? I hope you had fun navigating through the math maze!
To determine the distance the ship has sailed from the port, we need to use trigonometry and the given bearings.
First, let's draw a diagram to visualize the situation:
N
|
|
Lighthouse A -----------ship------- Port B
1. Determine the initial angle between Lighthouse A and Port B:
The bearing of Port B from Lighthouse A is given as N73E.
Since the bearing is measured clockwise from the north, we can subtract it from 90 degrees to get the included angle:
Initial angle = 90° - 73° = 17°
2. Determine the angle between the ship and Lighthouse A:
The bearing of the ship from Lighthouse A is given as N31E.
Again, subtracting it from 90 degrees, we get:
Angle between the ship and Lighthouse A = 90° - 31° = 59°
3. Determine the angle between the ship and Port B:
Since the ship has sailed due west from the port, the angle between the ship and Port B is simply the sum of the initial angle and the angle between the ship and Lighthouse A:
Angle between the ship and Port B = Initial angle + Angle between the ship and Lighthouse A
= 17° + 59°
= 76°
4. Use the law of cosines to find the distance the ship has sailed from the port:
In the triangle formed by Lighthouse A, Port B, and the ship's position, we can use the law of cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case:
- a is the distance from Lighthouse A to Port B (4.7 km),
- b is the distance the ship has sailed from the port (what we're trying to find),
- C is the angle between the ship and Port B (76°).
Plugging in the values:
b^2 = (4.7 km)^2 + (4.7 km)^2 - 2 * 4.7 km * 4.7 km * cos(76°)
Calculating this equation will give us the squared value of the distance traveled by the ship from the port.
Would you like me to calculate the actual numerical answer for the distance the ship has sailed from the port?
To find the distance the ship has sailed from the port, we can use the concept of bearing.
First, let's visualize the situation. Imagine a triangle with Lighthouse A at one corner, Port B at another corner, and the ship's current position as the third corner.
Since the bearing from the lighthouse to the port is N73E, we know that if we draw a line from the lighthouse to the port, it will make an angle of 73 degrees east of north (clockwise).
Next, the bearing of the ship from the lighthouse is N31E, which means that the line connecting the lighthouse to the ship will make an angle of 31 degrees east of north (clockwise).
Now, we can determine the angle between the ship's position and the line connecting the lighthouse to the port. To do this, we subtract the ship's bearing (N31E) from the bearing of the port (N73E):
73 - 31 = 42 degrees.
So, the angle between the ship's position and the line connecting the lighthouse to the port is 42 degrees.
Now, because the ship moved due west from the port, we can draw a line from the port to the ship that is perpendicular to the line connecting the lighthouse to the port.
Since the two lines are perpendicular, the triangle with the ship, port, and lighthouse forms a right triangle.
Using trigonometry, we can determine the length of the leg connecting the ship to the port.
In a right triangle, the tangent function relates the lengths of the sides to the angle opposite the side. In this case, the angle is 42 degrees, and we want to find the length of the side opposite the angle, which is the distance the ship has sailed from the port.
The tangent function is defined as:
tan(angle) = opposite / adjacent.
In this case, we have:
tan(42 degrees) = opposite / 4.7 km.
To find the length of the side opposite the angle, we rearrange the equation:
opposite = tan(42 degrees) * 4.7 km.
Using a calculator, we find that:
tan(42 degrees) ≈ 0.9004.
Substituting this value into the equation, we get:
opposite ≈ 0.9004 * 4.7 km.
Calculating the result, we find that the ship has sailed approximately:
opposite ≈ 4.23 km.
Therefore, the ship has sailed approximately 4.23 km from the port.