X is a point 2800 m East of Y. Due North of X is another point Z whose bearing from Y is N63°E. P is due North of Y and its bearing from X is N25°W. Calculate XZ and XP (give your answers to four significant figures)?
Just draw a diagram. All those right angles help a lot.
XZ/2800 = tan(90-63)°
2800/XP = cos(90-25)°
To solve this problem, we can use trigonometry and bearings to find the distances XZ and XP.
Given information:
- X is a point 2800 m East of Y.
- Point Z is due North of X and its bearing from Y is N63°E.
- Point P is due North of Y and its bearing from X is N25°W.
Step 1: Find the distance XZ.
To find XZ, we need to find the horizontal and vertical components of XZ using trigonometric ratios.
Horizontal component of XZ:
We know that X is 2800 m East of Y. Since the bearing of Z from Y is N63°E, the horizontal component of XZ is the same as the horizontal distance from X to Y.
The horizontal component of XZ = 2800 m.
Vertical component of XZ:
The vertical component of XZ is the same as the vertical difference between X and Z.
To find the vertical component, we need to calculate the distance traveled from X to Z. This can be done by splitting the bearing into its North and East components.
North component of Z from Y:
The bearing is N63°E, which means it is inclined at an angle of 63° to the East of North. Therefore, the North component of Z from Y can be found using the sine function:
North component = Distance * sin(63°)
North component = 2800 * sin(63°)
North component = 2486.96 m (rounded to four significant figures)
East component of Z from Y:
The bearing is N63°E, which means it is inclined at an angle of 63° to the East of North. Therefore, the East component of Z from Y can be found using the cosine function:
East component = Distance * cos(63°)
East component = 2800 * cos(63°)
East component = 1393.84 m (rounded to four significant figures)
Vertical component of XZ:
The vertical component of XZ is the same as the vertical difference between X and Z.
Vertical component = North component of Z from Y
Vertical component = 2486.96 m (rounded to four significant figures)
Using the horizontal and vertical components, we can now calculate the distance XZ using the Pythagorean theorem:
XZ = √(Horizontal component)^2 + (Vertical component)^2
= √(2800^2 + 2486.96^2)
= √(7840000 + 6184901.17)
= √14084901.17
= 3758.59 m (rounded to four significant figures)
Therefore, XZ is approximately 3758.59 m.
Step 2: Find the distance XP.
To find XP, we need to find the horizontal and vertical components of XP using trigonometric ratios.
Horizontal component of XP:
The horizontal component of XP is the same as the horizontal difference between X and P.
To find the horizontal component, we need to calculate the distance traveled from X to P. This can be done by splitting the bearing into its North and West components.
North component of P from X:
The bearing is N25°W, which means it is inclined at an angle of 25° to the West of North. Therefore, the North component of P from X can be found using the sine function:
North component = Distance * sin(25°)
North component = 2800 * sin(25°)
North component = 1205.59 m (rounded to four significant figures)
West component of P from X:
The bearing is N25°W, which means it is inclined at an angle of 25° to the West of North. Therefore, the West component of P from X can be found using the cosine function:
West component = Distance * cos(25°)
West component = 2800 * cos(25°)
West component = 2481.78 m (rounded to four significant figures)
Vertical component of XP:
The vertical component of XP is the same as the vertical distance from Y to P.
The vertical component of XP is equal to the horizontal component of P from X, which is 2481.78 m (rounded to four significant figures).
Using the horizontal and vertical components, we can now calculate the distance XP using the Pythagorean theorem:
XP = √(Horizontal component)^2 + (Vertical component)^2
= √(2481.78^2 + 2481.78^2)
= √(6158623.68 + 6158623.68)
= √12317247.36
= 3505.56 m (rounded to four significant figures)
Therefore, XP is approximately 3505.56 m.
To calculate XZ and XP, we'll make use of the concepts of bearings and right-angled triangles. Let's break down the problem step by step:
1. Let's start by visualizing the locations of the points X, Y, Z, and P. Based on the information given, we have the following layout:
Y ----> X ----> Z
|
|
V
P
2. Next, let's calculate the distance XZ. We know that X is 2800 m East of Y, and Z is located due North of X with a bearing of N63°E from Y.
To find XZ, we can imagine a right-angled triangle with the hypotenuse being XZ, the base being XY, and the altitude being YZ.
Using trigonometry, we can find XY and YZ using the given information and the right-angled triangle:
XY = 2800 m (given)
YZ = XY × tan(63°) [as tan(θ) = opposite/adjacent]
= 2800 m × tan(63°)
Now, we can calculate XZ using the Pythagorean theorem: XZ² = XY² + YZ²
XZ = √(XY² + YZ²)
= √(2800 m² + (2800 m × tan(63°))²)
= √(2800 m² + (2800 m × tan(63°))²)
Calculate XZ using the above expression, and round the answer to four significant figures.
3. Lastly, let's calculate XP. We know that P is due North of Y, and its bearing from X is N25°W.
Since P is directly due North of Y, the distance XP is equal to the distance XY.
Calculate XP as XY using the given value for XY, and round the answer to four significant figures.
By following these steps, you should now be able to calculate XZ and XP with the appropriate rounding.