A ship leaves port at 5 am and heads due east at 23 knots. At 10 am, to avoid a storm the ship changes course to N 60° E 60° east of north Find the ships distance from port at 4 pm. Round to the nearest tenth.
make a sketch
distance along first direction = 5 hrs x 23 knots
= 115 n miles
distance along 2nd direction = 6(23) = 138 n miles
angle between the two directions = 150°
by cosine law:
x^2 = 115^2 + 138^2 - 2(115)(138)cos 150°
= 59756.64...
x = 244.45 nautical miles
= appr 244.5 n miles
To find the ship's distance from port at 4 pm, we need to break the journey into two parts: from 5 am to 10 am, and from 10 am to 4 pm.
First, let's calculate the distance covered from 5 am to 10 am. The ship travels due east at a speed of 23 knots for 5 hours, so the distance covered is:
Distance1 = Speed1 * Time1
Distance1 = 23 knots * 5 hours = 115 nautical miles (rounded to the nearest whole number)
Next, let's calculate the distance covered from 10 am to 4 pm. The ship changes its course to N 60° E (60° east of north). This means the ship is heading 30° east of due north.
Using the given information, we know the ship is traveling at 23 knots from 10 am to 4 pm, so we can calculate the distance covered in this leg of the journey:
Distance2 = Speed2 * Time2
Distance2 = 23 knots * 6 hours = 138 nautical miles (rounded to the nearest whole number)
Now, we need to find the total distance from port at 4 pm. To do this, we can use the law of cosines to find the distance d between the two points, using the distances covered in each leg of the journey (Distance1 and Distance2):
d^2 = Distance1^2 + Distance2^2 - 2 * Distance1 * Distance2 * cos(30°)
Plugging in the values we calculated:
d^2 = 115^2 + 138^2 - 2 * 115 * 138 * cos(30°)
(Using 138.6 for √3/2 rounded to the nearest tenth)
d^2 = 13225 + 19177.96 - 2 * 115 * 138.6 * √3/2
d^2 = 13225 + 19177.96 - 2 * 115 * 138.6 * 0.866
d^2 = 13225 + 19177.96 - 2 * 115 * 120.1616
d^2 = 13225 + 19177.96 - 2 * 13839.04
d^2 = 13225 + 19177.96 - 27678.08
d^2 = 474.88
d = √474.88
d ≈ 21.8 nautical miles (rounded to the nearest tenth)
Therefore, the ship's distance from port at 4 pm is approximately 21.8 nautical miles.