A ship leaves port 21km on a bearing of 032 degree & 45km on a bearing of 287degrees
a, calculate it's distance from the port
b, calculate the bearing of the port from the ship
a) Well, if the ship left port, I guess it's not there anymore! So, we need to calculate the distance from the ship to the port. Let's use a bit of trigonometry here. We have two sides of a triangle, so we can use the Pythagorean theorem to find the remaining side. The first side is 21km, the second side is 45km. So, the distance from the ship to the port is √(21^2 + 45^2) = √(441 + 2025) = √2466 ≈ 49.66 km.
b) Now, let's calculate the bearing of the port from the ship. First, we need to find the angle between the ship's heading and the direction towards the port. We can do this by subtracting the bearing of the ship from 180 degrees. So, 180° - 032° = 148°. Therefore, the bearing of the port from the ship is 148 degrees. Keep in mind that compass bearings are usually expressed as 3-digit numbers, so we'd say it's 148 degrees.
To calculate the distance from the port, we can use the Pythagorean theorem and calculate the hypotenuse of a right triangle formed by the ship's movements.
a) Distance from the port:
We can calculate the horizontal and vertical distances traveled by the ship using trigonometric functions.
Horizontal distance:
To calculate the horizontal distance traveled, we can use the side adjacent to the angle of 032 degrees.
cos(032) = horizontal distance / 21km
Let's calculate the horizontal distance:
horizontal distance = cos(032) * 21km
Vertical distance:
To calculate the vertical distance traveled, we can use the side adjacent to the angle of 287 degrees.
sin(287) = vertical distance / 45km
Let's calculate the vertical distance:
vertical distance = sin(287) * 45km
Now, we can calculate the distance from the port (the hypotenuse of the right triangle) using the Pythagorean theorem:
distance from the port = √(horizontal distance^2 + vertical distance^2)
Let's substitute the values and calculate the distance from the port.
b) Bearing of the port from the ship:
To calculate the bearing of the port from the ship, we can use trigonometric functions and inverse trigonometric functions.
The bearing is the angle measured from the north direction in a clockwise direction. We can calculate the angle using the tangent function.
tan(bearing) = vertical distance / horizontal distance
Let's calculate the bearing of the port from the ship using the inverse tangent function:
bearing = atan(vertical distance / horizontal distance)
Now, let's substitute the values and calculate the bearing of the port from the ship.
To solve this problem, we can use the principles of trigonometry, specifically the cosine law and sine law. Let's break down the problem step by step:
a) To find the distance from the port, we can use the cosine law. The cosine law states that in a triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice the product of their lengths and the cosine of the included angle.
In this case, we have a triangle with two sides of lengths 21km and 45km, and the included angle is given as 287 - 32 = 255 degrees (bearing from the port back to the ship). Let's calculate the distance:
Let c be the distance from the port.
c^2 = 21^2 + 45^2 - 2 * 21 * 45 * cos(255 degrees)
To evaluate this equation, you may need to convert the angle to radians since most trigonometric functions accept radians. Remember that 360 degrees is equal to 2π radians.
After solving the equation, take the square root of both sides to find the distance c.
b) To find the bearing of the port from the ship, we can use the sine law. The sine law states that in a triangle, the ratio of a side's length to the sine of its opposite angle is the same for all sides and their opposite angles.
In this case, we have a triangle with sides of lengths 21km, 45km, and c (the distance from the port). We want to find the angle opposite the side of length c, which represents the bearing of the port from the ship.
Let A be the angle opposite the side of length 21km and B be the angle opposite the side of length 45km.
sin(A) / 21 = sin(B) / 45
Solve this equation for A, using the fact that sin^-1 is the inverse of the sine function.
Once you have the angle A, subtract it from 360 degrees to find the bearing of the port from the ship.
Remember to convert angles to the appropriate unit (degrees or radians) for your calculations and to use a calculator or math software to perform the trigonometric functions accurately.
Draw the vectors involved.
resolve each vector into its x- and y-components
(a) add them up and use the distance formula for the resultant.
(b) If the ship started at (0,0) and ended at (x,y), then the bearing of the port is θ (measured counterclockwise from the positive x-axis), such that
tanθ = y/x
Then convert that to a 0-360 bearing.