A ship leaves port at 7 am and heads due east at 34 knots. At 10 am, to avoid a storm the ship changes course to N 57° east of north). Find the ships distance from port at 2 pm. Round to the nearest tenth.
To determine nautical miles multiply the speed in knots by the number of hours.
(ex. 10knots in 2hrs would be;
10knots x 2hrs = 20 nautical miles.)
10am.-7am.= 3 hours due east from port..
at 10 am it changes direction to the vector specified above..so
2pm-10am= 4 hours from the point where it shifts direction
east of north means vehicle pointing north changes direction 57 deg.. to east..
so..3hrs. X 34 knots = 102naut. miles from port and due east..
and 4hrs X 34 knots= 136 naut miles. from the point of shifting direction..
if you draw it. it would form an oblique triangle..
with the angle for the unknown longer side x to be 147 degrees..
to solve for the longer side use cosine law..c^2=a^2+b^2-2abCosC
where c=147 degrees, a=102 mi. b=136 mi.
so the distance (inclined distance) of the ship from port is= 228.40 naut miles..
if you are asking the total distance travelled due east till it shifts then it would be 238 naut miles..
To solve this problem, we need to break it down into two parts: the ship's movement from 7 am to 10 am and its movement from 10 am to 2 pm.
1. From 7 am to 10 am:
During this period, the ship travels due east at a speed of 34 knots for 3 hours. The distance covered during this time is calculated by multiplying the speed in knots by the number of hours:
Distance = Speed x Time
Distance = 34 knots x 3 hours
Distance = 102 nautical miles
2. From 10 am to 2 pm:
At 10 am, the ship changes its course to N 57° east of north. To determine the ship's movement in this direction, we need to calculate the vertical and horizontal components separately.
Vertical component:
To find the vertical component, we multiply the speed by the cosine of the angle:
Vertical component = 34 knots * cos(57°)
Vertical component ≈ 34 knots * 0.5592
Vertical component ≈ 19.0368 knots
Horizontal component:
To determine the horizontal component, we multiply the speed by the sine of the angle:
Horizontal component = 34 knots * sin(57°)
Horizontal component ≈ 34 knots * 0.8290
Horizontal component ≈ 28.146 knots
Now we have the new direction and speed for the ship. From 10 am to 2 pm, the ship sails at a speed of 19.0368 knots in the vertical direction and 28.146 knots in the horizontal direction.
To determine the distance covered during this period, we multiply the speed in knots by the number of hours:
Distance = Speed x Time
Distance = √[(Vertical component)^2 + (Horizontal component)^2] x 4 hours
Distance = √[(19.0368 knots)^2 + (28.146 knots)^2] x 4 hours
Distance ≈ √[362.2846 + 791.9401] x 4
Distance ≈ √1154.2247 x 4
Distance ≈ 34.0043 knots x 4
Distance ≈ 136.0172 nautical miles
To find the total distance covered by the ship, we add the distance travelled in each segment:
Total distance = Distance from 7 am to 10 am + Distance from 10 am to 2 pm
Total distance = 102 nautical miles + 136.0172 nautical miles
Total distance ≈ 238.0172 nautical miles
Therefore, the ship's distance from the port at 2 pm is approximately 238.0 nautical miles when rounded to the nearest tenth.