Two ships A and B left a port each with a uniform speed on bearing 045 degree and x degree respectively. After 4hrs, they covered 100km and 120km respectively with B on a bearing of 105 degree from A. Calculate correct to 1 decimal place the value of x
Draw the diagram. You can see that the angle at A is 120°
You can find the angle at B using the law of sines:
sinB/100 = sin120°/120
Now, you know that
x-45 + 120 + B = 180
All angles are measured CW from +y-axis.
Given: PA = 100km[45o].
PB = 120km[Xo].
Bearing AB = 105o.
A = 105-45 = 60o = angle between PA and AB.
sin B/100 = sin60/120
B = 46.2 degrees.
P = 180-A-B = 180-60-46.2 = 73.8 degrees.
X = 45+P = 45+73.8 = 118.8 degrees.
To solve this problem, we can use the law of cosines for triangles. Let's break down the information given and solve step by step:
1. Ship A travels on a bearing of 045 degrees and covers a distance of 100 km in 4 hours.
2. Ship B travels on a bearing of x degrees and covers a distance of 120 km in 4 hours.
3. Ship B is on a bearing of 105 degrees from Ship A after 4 hours.
Let's find the angle between ships A and B using the law of cosines:
1. The distance between ships A and B after 4 hours is given by the equation: c^2 = a^2 + b^2 - 2ab * cos(C), where:
- a = 100 km (distance covered by Ship A)
- b = 120 km (distance covered by Ship B)
- c is the distance between Ship A and Ship B
- C is the angle between ships A and B
Substituting the values, we have: c^2 = 100^2 + 120^2 - 2 * 100 * 120 * cos(C)
2. We also know that the bearing between Ship A and Ship B after 4 hours is 105 degrees. In a triangle, the angle between the distance traveled by Ship A and the distance traveled by Ship B is equal to the bearing between the two ships. Therefore, C = 105 degrees.
3. Now we can plug in the values into the equation: c^2 = 100^2 + 120^2 - 2 * 100 * 120 * cos(105)
Calculating this, we find c^2 ≈ 10000 + 14400 - 24000 * cos(105)
4. We can simplify further: c^2 ≈ 24400 - 24000 * cos(105)
5. Taking the square root of both sides, we have c ≈ √(24400 - 24000 * cos(105))
Calculating this, we find c ≈ 14.3 km (rounded to 1 decimal place)
Therefore, the value of x (the bearing Ship B is traveling on) is approximately 14.3 degrees (rounded to 1 decimal place).
To solve this problem, we'll use the concept of relative motion. Let's break down the given information:
Ship A:
- Speed: Uniform speed (not mentioned in the question)
- Bearing: 045 degrees
- Distance covered: 100 km
- Time taken: 4 hours
Ship B:
- Speed: Uniform speed (not mentioned in the question)
- Bearing: x degrees
- Distance covered: 120 km
- Relative bearing from A: 105 degrees
First, let's find out the speed of both ships.
Ship A:
Speed = Distance covered / Time taken = 100 km / 4 hours = 25 km/h
Ship B:
Speed = Distance covered / Time taken = 120 km / 4 hours = 30 km/h
Now, let's consider the relative motion of Ship B with respect to Ship A.
The relative bearing of Ship B from Ship A is 105 degrees. Since Ship B is to the right of the bearing of Ship A, we subtract the bearing of Ship A from the relative bearing of Ship B.
Relative bearing = 105 degrees - 045 degrees = 60 degrees
Now we can apply trigonometry to find the angle x.
We have a triangle with two sides given: the adjacent side (120 km) and the hypotenuse (relative speed of Ship B with respect to Ship A).
We can use the cosine function: cos(x) = adjacent / hypotenuse.
cos(x) = 120 km / relative speed of Ship B with respect to Ship A
x = arccos(120 km / relative speed of Ship B with respect to Ship A)
Now, substitute the relative speed of B with respect to A:
x = arccos(120 km / (30 km/h - 25 km/h))
x ≈ arccos(24)
Using a calculator, we find x ≈ 75.5 degrees.
Therefore, the value of x is approximately 75.5 degrees (correct to 1 decimal place).