A boat sails on a bearing of N15°E for 10 km and then on a bearing of S85°E until it is due east of the starting point. What is the distance from the starting point to the nearest kilometre?
So, I got:
b/sin(85) = 10/sin(5)
b = 10sin(85)/sin(5)
b = 114.3km
Answer is 113 km
Can anyone tell me what I have done wrong? Thank you.
Your symbols lead to confusion. They are both incredibly archaic methods of direction notation. All modern navigation systems use north as a reference and measure degrees CW from that reference.
I can tell from your numbers you did not draw the vectors head to tail, and figure the angle between them correctly.
As bobpursley noted, you did not calculate the angles correctly.
you have to add 85+15 for the top angle.
My diagram has angles of 75, 100, and 5 degrees, and I get
b/sin100 = 10/sin5
b = 112.994.. or appr 113
To find the distance from the starting point to the nearest kilometer, we need to break down the boat's path and calculate the horizontal and vertical components separately.
First, let's understand the path of the boat:
1. The boat sails on a bearing of N15°E for 10 km.
2. Then, it sails on a bearing of S85°E until it is due east of the starting point.
To calculate the horizontal and vertical components, we need to use trigonometry. Let's start with the horizontal component:
1. The boat initially sails N15°E, which means it deviates 15° counterclockwise from the north axis.
2. The horizontal component can be calculated using cosine, where cos(15°) = adjacent/hypotenuse.
Let's call this distance x.
x = 10 km * cos(15°)
Next, let's calculate the vertical component:
1. The boat then changes its bearing to S85°E, which means it deviates 85° counterclockwise from the south axis.
2. The vertical component can be calculated using sine, where sin(85°) = opposite/hypotenuse.
Let's call this distance y.
y = 10 km * sin(85°)
The boat moves east until it is due east of the starting point. This means the vertical component (y) should cancel out the horizontal component (x).
Since the vertical component is negative (south direction) and the horizontal component is positive (east direction), we have the equation:
y = -x
Now, we can substitute the values of x and y:
-10 km * cos(15°) = 10 km * sin(85°)
By solving this equation, we can find the value of x:
cos(15°) = sin(85°) (Divide both sides by -10 km)
Now, using a scientific calculator:
cos(15°) ≈ 0.9659
sin(85°) ≈ 0.9997
Therefore, 0.9659 = 0.9997
This equation is not true, which means there might be a mistake in the problem or the given angles.
Please recheck the provided angles and make sure they are accurate, as the answer cannot be determined with the angles provided.