If a ship leaves port at 9:00 a.m. and sails due south for 3 hours at 14 knots, then turns N 60° E for another 2 hours, how far from port is the ship?
north = -3*14+2*14*cos 60
east = 2*14*sin 60
d^2 = north^2 + east^2
To calculate how far the ship is from the port, we need to break down the ship's movement into two components: its southward movement and its northeastward movement.
First, let's calculate the southward component. The ship sails due south at a speed of 14 knots for 3 hours. The formula to calculate distance is speed multiplied by time. So, the distance traveled southward can be calculated as 14 knots * 3 hours = 42 nautical miles.
Next, let's calculate the northeastward component. The ship changes its course to N 60° E and sails at the same speed of 14 knots for 2 hours. To calculate the distance in this direction, we need to decompose the speed into its x and y components. Given that the angle is 60°, the x component can be calculated using cosine, and the y component using sine.
x component = speed * cos(angle)
x component = 14 knots * cos(60°)
x component = 14 knots * 0.5
x component = 7 knots
y component = speed * sin(angle)
y component = 14 knots * sin(60°)
y component = 14 knots * (√3/2)
y component = 7 knots * √3
Since the ship traveled at a constant speed of 14 knots for 2 hours, the distance traveled northeastward is:
Northeastward distance = √((x component)^2 + (y component)^2) * time
Northeastward distance = √((7 knots)^2 + (7 knots * √3)^2) * 2 hours
To find the square of a value, we multiply it by itself. Thus:
Northeastward distance = √(49 knots^2 + (49 knots^2 * 3)) * 2 hours
Northeastward distance = √(49 knots^2 + 147 knots^2) * 2 hours
Northeastward distance = √(196 knots^2) * 2 hours
Northeastward distance = 14 knots * 2 hours
Northeastward distance = 28 nautical miles
Now, we can find the total distance from the port by using the Pythagorean theorem, which states that the square of the hypotenuse (total distance) is equal to the sum of the squares of the other two sides (southward distance and northeastward distance).
Total distance = √((southward distance)^2 + (northeastward distance)^2)
Total distance = √((42 nautical miles)^2 + (28 nautical miles)^2)
Total distance = √(1764 nautical miles^2 + 784 nautical miles^2)
Total distance = √(2548 nautical miles^2)
Total distance ≈ 50.48 nautical miles
Therefore, the ship is approximately 50.48 nautical miles away from the port.