A ship is sailing due north. At a certain point the bearing of the lighthouse 12.5 km away is N 38.8 degrees E. Later on, the captain notices that the bearing of the lighthouse has become S 44 degrees E. How far did the ship travel between the two observations of the lighthouse?
Did you make your sketch?
I have a triangle with angles 38.8° , 136°, and 5.2°
using the Sine Law:
If d is the distance travelled during the two observations ....
d/sin5.2 = 12.5/sin136
s = 12.5 sin 5.2/sin136= 1.63 km
Ahoy there! Well, it seems like the ship has taken quite the detour! Let's calculate how far it actually traveled.
First, let's break it down. The initial bearing of N 38.8 degrees E and the final bearing of S 44 degrees E means the ship turned by 90 degrees.
Now, since we have a right angle triangle formed by the ship, the lighthouse, and the distance between them, we can use the Pythagorean Theorem to find the distance traveled.
The distance traveled is equal to the hypotenuse of this triangle. Let's call it 'd'.
Using some math (I'm good at that too!), we can determine that sin(90°) = opposite/hypotenuse.
sin(90°) = sin(38.8° + 44°) = sin(82.8°) which is approximately 0.988.
So, we have 0.988 = 12.5 km / d.
Now, if we rearrange that equation and solve for 'd', we get d = 12.5 km / 0.988.
Calculating that, the ship traveled approximately 12.65 km between the two observations of the lighthouse. The ship sure knows how to take the scenic route, doesn't it?
To find the distance the ship traveled between the two observations of the lighthouse, we can use trigonometry.
Let's break down the problem into two parts:
1. First observation:
From the first observation, we know that the bearing of the lighthouse is N 38.8 degrees E. This means that the angle between the ship's direction (north) and the line connecting the ship to the lighthouse (the bearing) is 38.8 degrees.
Using trigonometry, we can find the distance between the ship and the lighthouse at the first observation point:
Distance = Opposite side / Tangent(angle)
Distance = 12.5 km / Tan(38.8°)
Calculating this, we get:
Distance = 12.5 km / Tan(38.8°) ≈ 12.5 km / 0.807 ≈ 15.48 km
So, at the first observation, the ship is approximately 15.48 km away from the lighthouse.
2. Second observation:
From the second observation, we know that the bearing of the lighthouse has become S 44 degrees E. This means that the angle between the ship's direction (south) and the line connecting the ship to the lighthouse (the bearing) is (180 - 44) = 136 degrees.
To find the distance between the ship and the lighthouse at the second observation point, we can use the same formula as before:
Distance = Opposite side / Tangent(angle)
Distance = x / Tan(136°) (where x is the distance we are trying to find)
For the second observation, we can rearrange the formula to solve for x:
x = Distance * Tan(angle)
Plugging in the values, we get:
x = 15.48 km * Tan(136°)
Calculating this, we get:
x = 15.48 km * (-2.59) ≈ -40.06 km (Note: Since the bearing is south, the distance is negative)
Therefore, between the two observations of the lighthouse, the ship traveled a distance of approximately 40.06 km.
To find the distance the ship traveled between the two observations of the lighthouse, we need to use trigonometry and the concept of bearings.
Let's break down the problem step by step:
1. Start by drawing a diagram. Draw a horizontal line to represent the shoreline, and mark a point on the line to represent the ship's initial position. Label this point as "Ship."
2. Draw a vertical line from the Ship point to represent the lighthouse. Label the top point as "Lighthouse."
3. From the Ship point, measure 12.5 km to the right (east) along the shoreline. Label this point as "Observation 1."
4. Using the given bearing of N 38.8 degrees E, draw a line from Observation 1 to the Lighthouse. This line represents the initial line of sight.
5. From the Ship point, measure a distance due south. Label the point where this line intersects the shoreline as "Observation 2."
6. Using the given bearing of S 44 degrees E, draw a line from Observation 2 to the Lighthouse. This line represents the second line of sight.
Note: At this point, you should have a triangle with the Lighthouse at the top, Observation 1 to the right, Observation 2 to the left, and the shoreline forming the base of the triangle.
Now let's find the distance the ship traveled between the two observations:
1. Notice that the two angles formed (38.8 degrees and 44 degrees) are both angles between the initial line of sight and the second line of sight.
2. Subtract these angles from 180 degrees to find the two angles formed inside the triangle. (180 - 38.8 = 141.2 degrees, and 180 - 44 = 136 degrees)
3. Identify which angle corresponds to the side you want to find (the distance the ship traveled). In this case, the angle of 136 degrees corresponds to the distance between Observation 1 and Observation 2.
4. Now, we can use the Law of Sines to find the length of the side corresponding to the angle of 136 degrees.
The Law of Sines states:
a/sin(A) = b/sin(B) = c/sin(C)
In our triangle, side b corresponds to angle B of 136 degrees, and side c corresponds to angle C of 38.8 degrees.
The formula becomes:
b/sin(B) = c/sin(C)
Substituting the known values:
b/sin(136) = 12.5 km/sin(38.8)
Rearranging the formula to solve for b:
b = sin(136) * (12.5 km/sin(38.8))
Calculate this value to find the length of side b, which represents the distance the ship traveled between the two observations.
Hence, the ship traveled approximately b kilometers between the two observations of the lighthouse.