An aeroplane flies for 4hours from position P on a bearing of N60W To poing Q At an average speed of 60km/hr. the aeroplane then flies on a bearing of 150 from Q To another point R 500km away. Calculate correct to 3 significant figures.
a) the distance /PR/
b) the bearing of R From P.
distance = speed * time
Draw a diagram. You can see that
(a) using the law of cosines,
PR^2 = 240^2 + 500^2 - 2*240*500 cos30°
(b) If P is at (0,0) then R is at (42.15,-313)
so the bearing of R from P is S7°W
Well, well, well! It seems like the aeroplane is going on a grand adventure! Let's help it out, shall we?
a) To find the distance /PR/, we need to break it down into smaller steps. First, we need to find the distance from P to Q, and then add the distance from Q to R.
The distance from P to Q is a bit of a trick question because we have the average speed (60 km/hr) and the time (4 hours). Ah, math tricks! To find distance, we simply multiply the speed by time: 60 km/hr * 4 hours = 240 km.
Now, for the distance from Q to R, we have a straightforward value of 500 km.
To find the total distance /PR/, we add the distances together: 240 km + 500 km = 740 km.
Therefore, the distance /PR/ is 740 km.
b) Ah, the bearing of R from P. What a directionally intriguing question! To find this, we need to use a bit of trigonometry. Are you ready?
Since the aeroplane is flying on a bearing of N60W from P to Q, and then turns to a bearing of 150 from Q to R, we can imagine a triangle (Delta PQR) where P is the vertex, and Q and R are the other two points. The angle at P is the sum of the two bearings from the problem: 60° + 150° = 210°.
Voila! The bearing of R from P is 210°.
Remember, though, that this is a humor-oriented response, so take the calculation with a grain of clown sprinkles!
To calculate the values asked:
a) Distance /PR/:
Since the aeroplane travels at an average speed of 60 km/hr for 4 hours, the distance covered can be calculated using the formula:
Distance = Speed x Time
Distance = 60 km/hr x 4 hours = 240 km
Therefore, the distance /PR/ is 240 km.
b) Bearing of R from P:
To find the bearing of R from P, we need to determine the angle formed between the direction from P to R and the north direction.
First, let's determine the angle formed between the direction from P to Q and the north direction:
The bearing of N60W means that the angle formed between the north direction and the line from P to Q is 60 degrees towards the west.
Next, let's determine the angle formed between the direction from Q to R and the direction from P to Q:
The bearing of 150 degrees means that the angle formed between the line from Q to R and the line from P to Q is 150 degrees.
To find the bearing of R from P, we need to add the two angles together:
Bearing of R from P = Angle between North and line from P to Q + Angle between line from Q to R and line from P to Q
Bearing of R from P = 60 degrees + 150 degrees = 210 degrees
Therefore, the bearing of R from P is 210 degrees.
To solve this problem, we can break it down into two parts: finding the distance PR and calculating the bearing of point R from point P.
a) To find the distance PR, we can use the information given about the average speed and time of flight. We know that the airplane flies for 4 hours at an average speed of 60 km/hr. Therefore, the distance covered from point P to point Q can be calculated using the formula:
Distance PQ = Average Speed × Time
Distance PQ = 60 km/hr × 4 hours
Distance PQ = 240 km
Now, we can calculate the distance PR by using the given information that point R is 500 km away from point Q and using the concept of Pythagoras' theorem. The distance PR forms the hypotenuse of a right-angled triangle, with the legs being PQ and QR.
Using Pythagoras' theorem:
PR² = PQ² + QR²
PR² = 240² + 500²
PR² = 57600 + 250000
PR² = 307600
Taking the square root of both sides:
PR ≈ √307600
PR ≈ 554.7 km (rounded to 3 significant figures)
b) To find the bearing of R from P, we need to use trigonometric functions. The bearing is the angle measured clockwise from the north, so we need to find the angle formed between the north direction and the line PR.
In a right-angled triangle, the angle formed between the hypotenuse and the adjacent side can be found using the inverse tangent (arctan) function. In this case, the adjacent side is PQ, and the opposite side is QR.
Using the arctan function:
tan(θ) = Opposite / Adjacent
tan(θ) = QR / PQ
tan(θ) = 500 / 240
θ = arctan(500 / 240)
θ ≈ 63.91 degrees
Since the bearing is measured clockwise from the north, the bearing of R from P is N63.91E (rounded to 3 significant figures).