An aeroplane flies from a town X on a bearing of N45 degree E to another town Y ,a distance of 200 cm it then changes course and flies to another town Z on a bearing of S60 degree E .if Z is directly tast of X .calculate ,correct to 3significant from X to Z
Draw triangle XYZ,
Law of sine: XY/sin45 = YZ/sin30,
200/sin45 = YZ/sin30,
200 = YZ*sin45/sin30,
YZ = 141.4 km.
200/sin45 = XZ/sinY,
200/sin45 = XZ/sin105,
XZ = ?.
To solve this problem, we can break it down into two components: the north-south displacement and the east-west displacement.
Step 1: Finding the north-south displacement
Since the airplane flies on a bearing of N45 degrees E from town X to Y, we can use the cosine function to find the north-south displacement.
cos(45°) = north-south displacement / 200 cm
north-south displacement = 200 cm * cos(45°)
Step 2: Finding the east-west displacement
Since the airplane changes course and flies on a bearing of S60 degrees E from town Y to Z, we can use the sine function to find the east-west displacement.
sin(60°) = east-west displacement / 200 cm
east-west displacement = 200 cm * sin(60°)
Step 3: Finding the total displacement from X to Z
To find the total displacement, we can add the north-south and east-west displacements.
Total displacement = square root of ((north-south displacement)^2 + (east-west displacement)^2)
Total displacement = square root of ((200 cm * cos(45°))^2 + (200 cm * sin(60°))^2)
Now, let's calculate the values:
cos(45°) = 0.707 (rounded to 3 decimal places)
sin(60°) = 0.866 (rounded to 3 decimal places)
north-south displacement = 200 cm * 0.707 ≈ 141.4 cm
east-west displacement = 200 cm * 0.866 ≈ 173.2 cm
Total displacement = square root of ((141.4 cm)^2 + (173.2 cm)^2) ≈ 222.713 cm
Therefore, the distance from town X to town Z, corrected to 3 significant figures, is approximately 222.713 cm.
To calculate the distance from town X to town Z, we can use the concept of vectors and trigonometry.
First, let's break down the information given:
1. The initial bearing of the airplane from town X to town Y is N45°E.
2. The distance from X to Y is 200 cm.
3. The airplane then changes course and flies to town Z on a bearing of S60°E.
4. Town Z is directly east (tast) of town X.
To calculate the distance from X to Z, we can determine the components of the distances traveled in the north and east direction separately.
From X to Y:
- The north component traveled can be calculated using the sine function: sin(45°) = opposite/hypotenuse.
- The east component traveled can be calculated using the cosine function: cos(45°) = adjacent/hypotenuse.
- Since we only know the overall distance (200 cm), we can use the Pythagorean theorem: hypotenuse^2 = opposite^2 + adjacent^2.
From Z to Y:
- The south component traveled can be calculated using the sine function: sin(60°) = opposite/hypotenuse.
- The east component traveled can be calculated using the cosine function: cos(60°) = adjacent/hypotenuse.
Now let's calculate the north and east components separately:
From X to Y:
- North component: sin(45°) = opposite/hypotenuse
=> sin(45°) = opposite/200
=> opposite = sin(45°) * 200
=> opposite ≈ 141.421 cm
- East component: cos(45°) = adjacent/hypotenuse
=> cos(45°) = adjacent/200
=> adjacent = cos(45°) * 200
=> adjacent ≈ 141.421 cm
From Z to Y:
- South component: sin(60°) = opposite/hypotenuse
=> sin(60°) = opposite/200
=> opposite = sin(60°) * 200
=> opposite ≈ 173.205 cm
- East component: cos(60°) = adjacent/hypotenuse
=> cos(60°) = adjacent/200
=> adjacent = cos(60°) * 200
=> adjacent ≈ 100 cm
To find the distance from X to Z, we need to add the north component from X to Y and subtract the south component from Z to Y:
Distance from X to Z = North component (X to Y) - South component (Z to Y)
= 141.421 cm - 173.205 cm (since Z is south of Y)
≈ -31.784 cm
Correcting this to 3 significant figures, the distance from X to Z is approximately -31.8 cm.
Note: This negative value indicates that town Z is actually west of town X. If the bearing is given correctly, then there may be an error in the question.
In google paste:
An aeroplane flies from town X on a bearing of N45'E to another town Y, a distance of 200km. It then changes course and flies to another town Z on a bearing of S60'E. If Z is directly east of X, calculate to 3sf, the distance from X to Z and the distance from Y to XZ.
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