A ship sails on a steady course bearing 106 degrees from A to B.If B is 76 nautical miles further east than A,find,to the nearest nautical mile,how far the ship has sailed?
☆CAH
CosQ = Adjacent
Hypotenuse
Cos16 = x
76
76 × Cos16 = x × 76
76
73 = x
ans : x = 73 nautical miles due
east
Well, of course the ship has sailed! It's a ship, not a boat on land. But to answer your question, let's break it down.
Since the ship is sailing on a steady course bearing 106 degrees, we can assume it's going in a straight line.
Since B is 76 nautical miles further east than A, we can use a bit of trigonometry to figure out the distance.
If we draw a triangle with A as the starting point, B as the ending point, and the ship's path as the hypotenuse, we can use the cosine rule to find the distance.
Using the cosine rule, we can say that:
c^2 = a^2 + b^2 - 2ab * cos(C),
where a and b are the sides of the triangle and C is the angle between them, which is 106 degrees.
In this case, a is the distance between A and the ship's point, while b is the distance between the ship's point and B, which is 76 nautical miles.
So we have:
c^2 = a^2 + (a + 76)^2 - 2a(a + 76) * cos(106).
Now, we solve for c to find the distance the ship has sailed. Without further calculations, I would say it's safe to approximate the answer to the nearest nautical mile.
To find the distance the ship has sailed, we can use the law of cosines. In this case, we have a triangle with one side of length 76 nautical miles, and the angle opposite that side is 106 degrees. Let "x" represent the distance the ship has sailed.
Using the law of cosines, we have:
x^2 = 76^2 + x^2 - 2*76*x*cos(106 degrees)
Simplifying the equation:
x^2 = 5776 + x^2 - 152*x*cos(106 degrees)
152*x*cos(106 degrees) = 5776
x = 5776 / (152*cos(106 degrees))
Using a calculator, we find:
x ≈ 56.1
Therefore, the ship has sailed approximately 56.1 nautical miles.
To find the distance the ship has sailed, you can use trigonometry. The given information provides the bearing and the distance difference from A to B. Here's how you can calculate the distance:
1. Draw a diagram: Sketch a diagram with point A representing the starting point of the ship and point B representing the destination.
2. Determine the angle: The ship sails on a steady course, so the bearing from A to B is 106 degrees. This means that the angle between the course and the north direction is 106 degrees.
3. Calculate the north-south and east-west components: To determine the north-south and east-west components of the sailing distance, use trigonometry. The north-south component can be calculated as follows:
North-South Component = Distance * sin(Bearing)
In this case, since we need to find the distance, we can rearrange the formula as follows:
Distance = North-South Component / sin(Bearing)
Given that the bearing is 106 degrees, the formula becomes:
Distance = North-South Component / sin(106)
4. Find the east-west component: The east-west component indicates the additional distance traveled east from point A to point B, which is given as 76 nautical miles.
5. Calculate the distance: Use the Pythagorean theorem to find the distance the ship has sailed:
Distance (sailed) = √(North-South Component^2 + East-West Component^2)
Substituting the values obtained from steps 3 and 4 into the formula, you can find the distance the ship has sailed. Rounding the answer to the nearest nautical mile will provide the final result.
looks like just simply
cos16° = 76/AB
AB = 76/cos16 =appr 79 n miles