An aircraft flies from a town P to another town Q 85km north of P. The aircraft then flies Eastward to another town T which is on a bearing of 53 degree from P if the average speed of the plane is 700km/hr calculate:
(I) PT
(ii) The total time taken for the journey
(i) use the law of cosines
(ii) time = distance/speed
(note: the total distance is not PT)
(I) Well, if the aircraft is flying at an average speed of 700 km/hr, we can assume it's in a rush! So, let's quickly calculate the distance between P and T.
First, we need to find the distance covered when flying Eastward. Since the aircraft is flying on a bearing of 53 degrees from P, we can say that it is forming a right triangle with PT as the hypotenuse. The given distance from P to Q is 85 km. Now, we need to find the distance from Q to T.
To find the distance covered Eastward, we can use a little math magic called trigonometry. The cosine of 53 degrees is equal to the adjacent side over the hypotenuse. In this case, the adjacent side is the distance from Q to T, and the hypotenuse is PT. So, we can write the equation as:
cos(53°) = Q T/ PT.
Solving for QT, we have QT = PT * cos(53°).
Now we know that the distance from P to T (PT) is equal to the distance from P to Q (85 km) times the cosine of 53 degrees. So, PT = 85 km * cos(53°).
(ii) To calculate the total time taken for the journey, we need to know the time it takes to cover the distance from P to Q and the time it takes to cover the distance from Q to T. Since we know the average speed of the plane is 700 km/hr, we can divide the distance from P to Q (85 km) by the average speed to find the time taken for that leg of the journey. Let's call it tPQ.
tPQ = 85 km / 700 km/hr.
Next, we need to find the time taken to cover the distance from Q to T. For that, we can divide the distance from Q to T (QT) by the average speed of the plane (700 km/hr). Let's call it tQT.
tQT = QT / 700 km/hr.
Finally, to find the total time taken for the journey, we add the time taken for each leg of the journey:
Total time = tPQ + tQT.
Now, let's get real and calculate the values!
An aircraft flew from to p to another q85km north of p. the aircraft then flew eastward from p. if the average speed of the plane is 700km 1.\PT/ the total time taken for the journey.
To find PT, we can use trigonometry. Let's draw a diagram:
```
Q
|
|
| 85 km
|
P---T
```
We want to find PT. We also know that the angle PQT is 53 degrees. We can use the sine function to solve for PT:
sin(53) = PT / 85
PT = 85 * sin(53)
PT ≈ 68.2 km
To find the total time taken for the journey, we need to know the distance traveled Eastward from P. We can use the cosine function to find this distance:
cos(53) = distance East / PT
distance East = PT * cos(53)
distance East ≈ 49.7 km
The total distance traveled is the sum of the distances traveled North and East:
total distance = 85 + 49.7
total distance ≈ 134.7 km
Finally, we can use the formula time = distance / speed to find the total time taken:
time = total distance / speed
time = 134.7 / 700
time ≈ 0.193 hours or 11.6 minutes (rounded to one decimal place)
To calculate the values, we can use trigonometry.
Let's break down the given information step by step:
(I) PT:
1. Draw a diagram to represent the situation. Place points P, Q, and T on the diagram.
P______ T
| |
| |
| |
Q________
2. Since Q is 85km north of P, draw a line 85km long vertically upwards from point P to point Q.
P______ T
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| |
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Q________
3. The aircraft then flies eastward towards town T, which is on a bearing of 53 degrees from P. A bearing is the angle measured clockwise from the north direction.
P______ T
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| / /
| / /
Q________
4. To find PT, we need to find the horizontal distance from Q to T. To do this, we can use trigonometry.
5. Calculate the horizontal distance (PT) using the formula: PT = QP * cos(bearing), where bearing is 53 degrees.
PT = 85 km * cos(53 degrees)
PT = 85 km * 0.6
PT ≈ 51 km
Therefore, PT is approximately 51 km.
(ii) The total time taken for the journey:
To calculate the total time taken for the journey, we need the distance and speed of the plane.
1. The distance from P to Q is given as 85 km.
2. The distance from Q to T (PT) is approximately 51 km, as calculated in (I).
3. The average speed of the plane is given as 700 km/hr.
4. To calculate the total time, we need to find the time taken for each leg of the journey and then add them together.
5. The time taken for the leg from P to Q is obtained by dividing the distance by the speed:
Time taken for PQ = Distance / Speed
Time taken for PQ = 85 km / 700 km/hr
Time taken for PQ ≈ 0.12 hr
6. The time taken for the leg from Q to T is obtained by dividing the distance by the speed:
Time taken for QT = Distance / Speed
Time taken for QT = 51 km / 700 km/hr
Time taken for QT ≈ 0.07 hr
7. Now, add the times for both legs to find the total time:
Total time taken = Time taken for PQ + Time taken for QT
Total time taken ≈ 0.12 hr + 0.07 hr
Total time taken ≈ 0.19 hr
Therefore, the total time taken for the journey is approximately 0.19 hours (or 11 minutes and 24 seconds).