A ship leaves port at 6 am and heads due east at 20 knots. At 10 am, to avoid a storm the ship changes course to N 47° E.. (47° east of north). Find the ships distance from port at 4 pm.
first path:
4 hrs at 20 knots = 80 n miles
second path:
6 hrs at 20 knots = 120 n miles
angle between them = 90+47 = 137°
I see a clear case of the cosine law:
let x be the distance from port
x^2 = 80^2 + 120^2 - 2(80)(120)cos137
= ....
x = √....
you do the button-pushing
Sorry what do you mean by ^ ?
186.66 miles
To find the distance from the port at 4 pm, we need to break down the ship's journey into two separate segments: 1) from 6 am to 10 am and 2) from 10 am to 4 pm.
1) From 6 am to 10 am:
The ship travels due east at a speed of 20 knots for a total of 4 hours. Since the speed is given in knots and the time is given in hours, we can find the distance using the formula: Distance = Speed x Time.
Distance = 20 knots x 4 hours = 80 nautical miles.
2) From 10 am to 4 pm:
The ship changes its course to N 47° E. This means it is heading in a direction that is 47° east of north. To calculate the ship's distance from the port, we need to use trigonometry.
First, we need to find the distance the ship has traveled north from the port. Since the ship changed direction for 6 hours (from 10 am to 4 pm), we can calculate the northward distance using the formula: Distance north = Speed x Time = 20 knots x 6 hours = 120 nautical miles.
Next, we need to find the distance the ship has traveled east from the port. To do this, we can use trigonometric ratios. In this case, the given angle is 43° (90° - 47°). We can use the cosine of the angle to find the distance east. The formula is: Distance east = Distance north x Cosine(angle).
Distance east = 120 nautical miles x Cosine(43°) ≈ 86.6 nautical miles.
Finally, to find the ship's total distance from the port at 4 pm, we can use the Pythagorean theorem because the northward and eastward distances form a right triangle. The formula is: Distance = Square Root of (Distance north squared + Distance east squared).
Distance = Square Root(120^2 + 86.6^2) ≈ 147.2 nautical miles.
Therefore, the ship is approximately 147.2 nautical miles away from the port at 4 pm.