A man travel from a village x on a bearing of 060 to a village y he travel to a village z on a bearing of 195 if 9 is directly east of x. Calculate correct to third significant fiqure the distance (1) x from z (2) z from y
If you plot the triangle XYZ, you can see that angle Y = 45° and angle Z is 105°.
I have no idea what "9 is directly east of x" means, and since you have not given any actual distances, it is impossible to answer the questions.
But once you have at least one side of XYZ, you can use the law of sines or cosines to get the desired distances.
To calculate the distance between the villages, we will use the trigonometric formulas for finding the lengths of the sides of a triangle.
(1) Distance from X to Z:
Given that 9 is directly east of X, we can consider it as the base of a right-angled triangle.
Using trigonometry, we can find the length of the adjacent side (let's call it d) using the cosine function:
cos(60°) = adj/hypotenuse
cos(60°) = 9/d
d = 9 / cos(60°)
Calculating the value of d:
d = 9 / 0.5
d = 18
So, the distance from X to Z is 18 units.
(2) Distance from Z to Y:
We have the bearing from Z to Y as 195°, which is measured counterclockwise from the east direction. Since we already know the distance from X to Z (18 units), we can consider it as the base of another right-angled triangle.
To find the opposite side (let's call it x), we can use the sine function:
sin(195°) = opp/hypotenuse
sin(195°) = x/18
x = 18 * sin(195°)
Calculating the value of x:
x = 18 * (-0.423)
x = -7.6 (rounded to one decimal place)
Since distance cannot be negative, we take the magnitude of x:
| x | = 7.6
So, the distance from Z to Y is 7.6 units (rounded to one decimal place).
To solve this problem, we can break it down into simple steps:
Step 1: Understand the bearings and directions:
- A bearing is a direction measured in degrees from the north in a clockwise direction.
- A bearing of 060 means the man traveled 60 degrees clockwise from the north, which is slightly east of the east direction.
- A bearing of 195 means the man traveled 195 degrees clockwise from the north, which is mostly south, but slightly west of the south direction.
- Knowing that 9 is directly east of x, we can assume that 9 is a distance measurement in the east direction.
Step 2: Draw a diagram:
Let's draw a diagram to visualize the problem. Since we only need the distance, the exact scale is not necessary. We can assume any convenient scale where 1 unit represents 1 kilometer.
y
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----x ------ 9 ----z
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south
Step 3: Calculate the distances:
We need to find the distance between x and z and between z and y.
(1) Distance between x and z:
To calculate this distance, we can use trigonometry. We have a right-angled triangle with the hypotenuse being the distance between x and z, the side opposite the angle 060 as the east direction distance, and the side adjacent to the angle 060 as the north direction distance.
Using trigonometry, we can use the sine function to find the north direction distance (opposite side):
sin(60°) = north distance / hypotenuse
north distance = hypotenuse * sin(60°)
Since we know that 9 is directly east of x, the east direction distance is 9 km.
Now, we can calculate the hypotenuse (distance between x and z):
cos(60°) = east distance / hypotenuse
hypotenuse = east distance / cos(60°)
north distance = hypotenuse * sin(60°)
east distance = 9 km
cos(60°) = 0.5
hypotenuse = 9 km / 0.5
north distance = (9 km / 0.5) * sin(60°)
Calculating this, we get:
north distance = (18 km) * 0.866 = 15.588 km (rounding to 3 significant figures)
(2) Distance between z and y:
The distance between z and y can be calculated in the same way using trigonometry.
Using sine function:
sin(15°) = north distance / hypotenuse
north distance = hypotenuse * sin(15°)
Therefore:
north distance = (15.588 km) * sin(15°)
Calculating this, we get:
north distance = (15.588 km) * 0.259 = 4.043 km (rounding to 3 significant figures)
So, the answers are:
(1) The distance between x and z is approximately 15.588 km.
(2) The distance between z and y is approximately 4.043 km.