A ship P sails at an average speed of 50km/hr on a bearing of N55 degree E from a point R.At the same time another ship Q sails at an average speed of 40km/hr on a bearing of S25 degreeW.How far are they apart after 4hours
assuming they both left from P at the same time ....
you have a clear case of the cosine law.
x^2 = 200^2 + 160^2 - 2(200)(160)cos150°
Careful with the cos150° , it will be negative
All angles are measured CW from +Y-axis,
Dp = 50km/hr[55o] * 4hr. = 200km[55o] = Distance of ship P,
Dq = 40km/hr[205o] * 4hr. = 160km[205o] = Distance of ship Q.
D = 200km[55o] - 160km[205o],
X = 200*sin55 - 160*sin205 = 231.4 km,
Y = 200*Cos55 - 160*Cos205 = 259.7 km,
D = Sqrt(X^2+Y^2) = 347.8 km. = Distance between P and Q after 4 hrs.
Tan A = X/Y.
To find the distance between ships P and Q after 4 hours, we can use the distance formula in trigonometry.
First, let's calculate the positions of P and Q after 4 hours.
Ship P sails at an average speed of 50 km/hr on a bearing of N55°E. The bearing N55°E means that the ship is moving 55° east of north.
To calculate the position of ship P after 4 hours, we can use the formula:
distance = average speed × time
So, the distance sailed by ship P after 4 hours is:
distance_P = 50 km/hr × 4 hr = 200 km
Now, let's look at ship Q. It sails at an average speed of 40 km/hr on a bearing of S25°W. The bearing S25°W means that the ship is moving 25° west of south.
To calculate the position of ship Q after 4 hours, we can use the formula:
distance = average speed × time
So, the distance sailed by ship Q after 4 hours is:
distance_Q = 40 km/hr × 4 hr = 160 km
Now we have the positions of ships P and Q after 4 hours.
To find the distance between them, we can use the Law of Cosines. The formula is:
c² = a² + b² - 2abcos(C)
Where c is the distance between the ships, a and b are the distances sailed by ships P and Q, and C is the angle between the paths of the ships.
In this case, a = distance_P = 200 km, b = distance_Q = 160 km, and C is the angle that is formed between the paths of ships P and Q.
C can be found by subtracting the bearings of the ships from 180°.
C = (180° - 55°) + (180° - 25°) = 155°
Now, we can plug in the values into the formula:
c² = 200² + 160² - 2 × 200 × 160 × cos(155°)
Simplifying the equation:
c² = 40000 + 25600 - 64000 × cos(155°)
c² = 65600 - 64000 × cos(155°)
Using a calculator to calculate cos(155°), we find that cos(155°) ≈ -0.57358
Plugging in the value:
c² = 65600 - 64000 × (-0.57358)
c² = 65600 + 36626.56
c² = 102226.56
Taking the square root of both sides:
c ≈ √102226.56
c ≈ 319.53 km
Therefore, after 4 hours, ships P and Q are approximately 319.53 km apart.