Shopping centres X, Y and Z are such that Y is 12km south of X and Z is 15km from X, Z is on a bearing 330 from Y. Calculate the bearing of Z from X.
ZX^2 = 15^2 + 12^2 - 2*15*12cos30°
ZX = 7.565 km
Now, in triangle XYZ, we can find angle X using the law of sines
sinX/15 = sin30°/7.565
Now the bearing of Z from X is 270+(∡X-90)
Given: XY = 12km[180o], YZ = ___km[330o], XZ = 15km[??].
sinZ/12 = sin30/15.
Z = 24o.
330-270 = 60o
Y = 90-60 = 30o.
X = 180-24-30 = 126o.
YZ/sin126 = 15/sin30.
YZ = 24 km.
YZ = 24km[330o].
XZ = XY+YZ = 12[180o]+24[330o].
XZ = (12*sin180+24*sin330)+(12*cos180+24*cos330)i
XZ = -12 + 9i = 15km[-53o] = 15km[307o] CW.
To calculate the bearing of Z from X, we need to find the angle formed by the line connecting X and Z with respect to the north direction.
First, let's draw a diagram to help visualize the situation:
Y
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X -------------- Z
We are given that Y is 12km south of X. This means that the distance from X to Y on the vertical line is 12km.
We are also given that Z is 15km from X and is on a bearing of 330 from Y. A bearing is the angle measured clockwise from the north direction.
To find the bearing of Z from X, we can use trigonometry. Since we know the distances involved, we can use the tangent function to calculate the angle.
Let's define the angle formed by the line connecting X and Z as θ.
tan(θ) = (distance from Y to Z) / (distance from X to Y)
In this case, the distance from Y to Z is 15km, and the distance from X to Y is 12km.
tan(θ) = 15 / 12
Now we can solve for θ by taking the inverse tangent of both sides:
θ = arctan(15 / 12)
Using a calculator or a table of trigonometric values, we find that arctan(15 / 12) ≈ 49.18 degrees.
Therefore, the bearing of Z from X is approximately 49.18 degrees.
To solve this problem, we can use trigonometry and bearings.
Step 1: Draw a diagram
Draw a diagram that represents the given information. Place three points: X, Y, and Z. Label X as the starting point.
Step 2: Determine the distances
According to the information given, Y is 12km south of X, and Z is 15km from X. Draw lines to represent these distances on the diagram.
Step 3: Use bearings to identify angles
The question states that Z is on a bearing of 330 from Y. To understand this, imagine a compass with 360 degrees, where 0 degrees is North, 90 degrees is East, 180 degrees is South, and 270 degrees is West. A bearing of 330 means that Z is located in the West-Southwest direction. Mark this direction on the diagram.
Step 4: Determine the bearing of Z from X
Now, we want to find the bearing from X to Z. To do this, we need to find the angle between the line from X to Z and the north direction.
First, draw a line from X to Z on the diagram. Next, draw a line parallel to the North direction, intersecting the line from X to Z.
Step 5: Measure the angle
Using a protractor, measure the angle between the line from X to Z and the parallel line representing North. This angle represents the bearing of Z from X.
Step 6: Calculate the bearing
Once you have measured the angle, you can use it to calculate the actual bearing.
For example, let's say the angle measures 50 degrees. Since the North direction is 0 degrees, and we are measuring the angle clockwise, the bearing of Z from X would be 360 degrees (North) - 50 degrees = 310 degrees.
So, the bearing of Z from X is 310 degrees.
Note: The exact measurements and angle may differ depending on your specific diagram and measurements.