The bearing of ships A and B from a port P are 225 degree and 116 degree respectively.ship A is 3.9 km from ship B on a bearing of 258 degree. Calculate the distance of ship A from P
Please solve this with the aid of diagram
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https://www.jiskha.com/questions/1795272/The-bearings-of-ships-A-andB-from-Port-P-are-225-and-116-respectively-Ship-A
To solve this problem, we can use the concept of trigonometry and the properties of bearings.
Let's break down the information provided:
1. The bearing of ship A from port P is 225 degrees.
2. The bearing of ship B from port P is 116 degrees.
3. Ship A is 3.9 km from ship B on a bearing of 258 degrees.
To find the distance of ship A from port P, we can use the Law of Sines.
First, let's label the angles and sides of the triangle:
- Angle PAB: 180° - Bearing of A from P = 180° - 225° = 45°
- Angle PBA: 180° - Bearing of B from P = 180° - 116° = 64°
- Angle ABP: 180° - 258° = 78°
We'll label the sides as follows:
- Side PA: the distance from Ship A to Port P (what we want to calculate)
- Side PB: the distance from Ship B to Port P (also what we ultimately want to calculate)
- Side AB: the distance from Ship A to Ship B, which is given as 3.9 km
Now, we can use the Law of Sines:
sin(PAB) / PA = sin(PBA) / PB = sin(ABP) / AB
Plugging in the known values:
- sin(45°) / PA = sin(64°) / PB = sin(78°) / 3.9
We can rearrange the equation to solve for PA, the distance of ship A from port P:
PA = sin(45°) / (sin(64°) / PB) = (sin(45°) * PB) / sin(64°)
Now, we can substitute the known values and solve for PA:
PA = (sin(45°) * PB) / sin(64°)
To find PB, we can use the given information that Ship A is 3.9 km from Ship B on a bearing of 258 degrees. We can find PB using the Law of Cosines:
PB^2 = PA^2 + AB^2 - 2 * PA * AB * cos(258°)
Substituting the known values:
PB^2 = PA^2 + (3.9 km)^2 - 2 * PA * 3.9 km * cos(258°)
Now, let's solve for PB using this equation.
Once we have the value of PB, we can substitute it back into the equation for PA to find the distance of Ship A from Port P.
All given angles are measured CW from +Y-axis(0o).
sin(64+45)/3.9 = sin26/PA.
PA*sin109 = 3.9*sin26,
PA = 3.9*sin26/sin109 = 1.81 km.