The bearings of ships A andB from Port P are 225° and 116° respectively.Ship A is 3.9km from ship B on a bearing of 258°.Calculate the distance of ship A from P
Draw the triangle PAB
∡B = 38°
∡A = 33°
side p = 3.9
So, ∡P = 109°
Using the law of sines,
b/sin38° = 3.9/sin109°
b = 2.54 km
We are not going to calculate this for you. Get a life or drop out.
To calculate the distance of ship A from Port P, we can use trigonometry and the given information.
1. Start by drawing a diagram to represent the given information. The diagram should have Port P as the reference point, ship A, ship B, and the respective bearings and distances mentioned.
2. Label the unknown distance from Port P to ship A as "x."
3. From the given information, we can see that ship A is 3.9 km away from ship B on a bearing of 258°. This means we can draw a triangle with sides "x" (from Port P to ship A), 3.9 km (from ship A to ship B), and an angle of 12° (180° - 258°, as the sum of all angles in a triangle is 180°).
4. Next, we can use the law of cosines to find the value of "x." The law of cosines states that, in a triangle with sides of lengths a, b, and c, and an angle opposite side c (C), the following equation holds:
c^2 = a^2 + b^2 - 2*a*b*cos(C)
In our case, we can substitute the values as follows:
(3.9 km)^2 = x^2 + (3.9 km)^2 - 2*x*(3.9 km)*cos(12°)
Simplifying the equation gives:
15.21 km^2 = x^2 + 15.21 km^2 - 7.8 km*x*cos(12°)
Cancelling out the common terms on both sides yields:
0 = x^2 - 7.8 km*x*cos(12°)
Rearranging the equation gives:
x^2 - 7.8 km*x*cos(12°) = 0
5. Now, we can solve this quadratic equation for "x." There are several methods to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula. In this case, it is easier to use factoring.
The equation can be rewritten as:
x*(x - 7.8 km*cos(12°)) = 0
Setting each factor equal to zero gives two possible solutions:
x = 0 (which is not possible in this context, as it represents no distance) or
x - 7.8 km*cos(12°) = 0
Simplifying the equation further:
x = 7.8 km*cos(12°)
6. Finally, we can calculate the value of "x" using a calculator. Multiply 7.8 km by the cosine of 12°:
x ≈ 7.8 km * cos(12°) ≈ 7.8 km * 0.978 ≈ 7.6284 km
Therefore, ship A is approximately 7.6284 km away from Port P.