A ship leaves port and sails at a bearing of 124degrees. Another ship leaves the same port at the same time sailing at a bearing of 74degrees. When both ships are 80 miles from the port, how far are they from each other?
use the law of cosines. The included angle is 50°
To find the distance between the two ships, we need to use trigonometry and the concept of vector addition. Let's break down the problem step by step:
Step 1: Draw a diagram:
Draw a rough diagram representing the situation. Place the port at the origin (0,0) and label the two ships with their respective angles and distances from the port.
Step 2: Determine the positions of the ships:
Let's assume the coordinates of the first ship are (x₁, y₁) and the second ship are (x₂, y₂). Since both ships are 80 miles away from the port, their distance from the port can be used as the magnitude of the vectors.
For the first ship, we know:
x₁ = 80 * cos(124°)
y₁ = 80 * sin(124°)
For the second ship, we know:
x₂ = 80 * cos(74°)
y₂ = 80 * sin(74°)
Step 3: Find the distance between the ships:
To find the distance between the ships, we need to calculate the magnitude of the vector connecting their positions. The formula for magnitude is:
distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Substituting the coordinates we obtained earlier, we have:
distance = √(((80 * cos(74°)) - (80 * cos(124°)))² + ((80 * sin(74°)) - (80 * sin(124°)))²)
Simplifying the equation gives us the distance between the two ships when they are 80 miles from the port.