Two ships left the same port at the same time. One travelled at 16kn/h on a course of 275 degrees while the other at 20 km/h of 225 degrees. How far apart are the ships after a. 2 hours? b. 4 hours?
*note: this is a trigonometry question using the cosine law.
Thanks!
To solve this problem, we can use the cosine law to find the distance between the two ships. The cosine law states that in a triangle, with sides a, b, and c and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
Let's calculate the distance between the two ships after 2 hours and 4 hours:
a) After 2 hours:
Ship 1 travels at 16 km/h for 2 hours, covering a distance of d1 = 16 km/h * 2 hours = 32 km.
Ship 2 travels at 20 km/h for 2 hours, covering a distance of d2 = 20 km/h * 2 hours = 40 km.
Using the cosine law, the distance between the two ships after 2 hours is:
c^2 = 32^2 + 40^2 - 2 * 32 * 40 * cos(225 degrees)
To calculate the distance, we need to convert the angle to radians:
cos(225 degrees) is equal to cos(225 * (π/180)).
b) After 4 hours:
Ship 1 travels at 16 km/h for 4 hours, covering a distance of d1 = 16 km/h * 4 hours = 64 km.
Ship 2 travels at 20 km/h for 4 hours, covering a distance of d2 = 20 km/h * 4 hours = 80 km.
Using the cosine law, the distance between the two ships after 4 hours is:
c^2 = 64^2 + 80^2 - 2 * 64 * 80 * cos(225 degrees)
Again, we need to convert the angle to radians:
cos(225 degrees) is equal to cos(225 * (π/180)).
Now, let's calculate the distances between the ships after 2 hours and 4 hours.
To solve this problem, we will use the cosine law, which states that for any triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice the product of those sides by the cosine of the included angle.
a) After 2 hours:
Let's first calculate the distance traveled by each ship. Ship 1 is traveling at a speed of 16 km/h for 2 hours, so it has traveled a distance of 16 km/h * 2 h = 32 km. Similarly, Ship 2 is traveling at a speed of 20 km/h for 2 hours, so it has traveled a distance of 20 km/h * 2 h = 40 km.
Now, we can use the cosine law to find the distance between the two ships. We know the lengths of the two sides and the included angle. Let's call the distance between the two ships "d".
Using the cosine law:
d^2 = (32 km)^2 + (40 km)^2 - 2 * 32 km * 40 km * cos(275 - 225) degrees
Now, we need to convert the angle from degrees to radians:
275 degrees = (275 * π) / 180 radians
225 degrees = (225 * π) / 180 radians
Substituting the values:
d^2 = (32 km)^2 + (40 km)^2 - 2 * 32 km * 40 km * cos((275 * π) / 180 - (225 * π) / 180) radians
Using a calculator to simplify the expression, we can find the value of d.
b) After 4 hours:
Using the same approach as above, we can calculate the distance traveled by each ship and then use the cosine law to find the new distance between the ships.
Remember to adjust the time and angles accordingly.
275 - 225 = 50 degrees difference in heading
In two hours
Ship A went 16 * 2 = 32
Sbip B went 20 * 2 = 40
c^2 = a^2 + b^2 - 2 a b cos C
c^2 = (32)^2 + (40)^2 -2(32)(40)cos 50
c^2 = 1024 + 1600 - 1646.5
c^2 = 978.4
c = 31.3
For pert B, similar triangle, double time, double distance
= 62.6