The bearing of 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
2.54km
Draw the sketch. You have two angles (<BpA, <ApB, and the side opposite.
Law of Sines applies.
x/sin<pBA = 3.9/sin<apB
To calculate the distance of ship A from port P, we need to use the trigonometric concept of the sine rule.
Step 1: Draw a diagram with Port P at the center. Label the positions of Ship A and Ship B accordingly.
Step 2: The bearing of Ship A from Port P is given as 225 degrees. To calculate the angle PA𝜃, subtract 225 degrees from 360 degrees:
PA𝜃 = 360° - 225° = 135°
Step 3: The bearing of Ship B from Port P is given as 116 degrees. To calculate the angle PB𝜃, subtract 116 degrees from 360 degrees:
PB𝜃 = 360° - 116° = 244°
Step 4: The bearing of Ship A from Ship B is given as 258 degrees. The angle AB𝜃 can be calculated by subtracting 258 degrees from 360 degrees:
AB𝜃 = 360° - 258° = 102°
Step 5: Using the sine rule, we can set up the following equation:
Sin(PA𝜃) / Sin(PB𝜃) = AB / BP
Step 6: Since we know AB (distance between Ship A and Ship B) is 3.9 km, we can rewrite the equation as:
Sin(135°) / Sin(244°) = 3.9 km / BP
Step 7: Solve for BP by cross-multiplying and rearranging the equation:
BP = (3.9 km * Sin(244°)) / Sin(135°)
Step 8: Calculate the value of BP using a scientific calculator:
BP ≈ (3.9 km * -0.602) / 0.707 ≈ -3.33 km
The distance of Ship A from Port P (AP) is the absolute value of BP:
AP ≈ |-3.33 km| ≈ 3.33 km
Therefore, the distance of Ship A from Port P is approximately 3.33 kilometers.
To find the distance of ship A from port P, we can use trigonometry and the given information.
Step 1: Draw a diagram to represent the situation. Label port P, ship A, and ship B.
Step 2: From the given information, we know that the bearing of ship A from port P is 225 degrees and the bearing of ship B from port P is 116 degrees.
Step 3: Convert the bearings to angles measured from the positive x-axis (east), which is commonly used in mathematics. To do this, subtract the given angles from 360 degrees.
The angle of ship A from the positive x-axis = 360° - 225° = 135°
The angle of ship B from the positive x-axis = 360° - 116° = 244°
Step 4: Calculate the angle between ship B and ship A. To do this, subtract the angle of ship A from the angle of ship B.
Angle between ship B and ship A = 244° - 135° = 109°
Step 5: Use the Law of Cosines to find the distance of ship A from port P. The Law of Cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where c is the side opposite angle C and a and b are the lengths of the other two sides.
We want to find the distance of ship A from port P, which we will label as side a. Ship B is 3.9 km away from ship A, and we will label this distance as side b. The angle between ship B and ship A is 109°, which we will label as angle C.
Let's substitute the values into the formula:
a^2 = b^2 + c^2 - 2bc * cos(C)
a^2 = (3.9 km)^2 + (Distance of ship A from port P)^2 - 2 * 3.9 km * (Distance of ship A from port P) * cos(109°)
Step 6: Solve the equation for a. Rearrange the equation to solve for the distance of ship A from port P.
a^2 - (Distance of ship A from port P)^2 = (3.9 km)^2 - 2 * 3.9 km * (Distance of ship A from port P) * cos(109°)
(Distance of ship A from port P)^2 + 2 * 3.9 km * (Distance of ship A from port P) * cos(109°) - (3.9 km)^2 = 0
This is now a quadratic equation in terms of the distance of ship A from port P. You can solve this equation by substituting the values and solving for the distance of ship A from port P using the quadratic formula or any other appropriate method.
Once you have found the value for the distance of ship A from port P, you can calculate the distance.