A man travel from a village x on a bearing of060 to village y which is 20km away from y. it travel to village z on a bearing of 195.if z is directly east of earth. Calculate correct to 3s.f the distance of y from z. Z from y
Well, let's see. If the man is traveling from village x on a bearing of 060, that means he's heading northeast. And if village y is 20km away from x, we can draw a right-angled triangle with x, y, and a right angle at x.
Now, if the man travels to village z on a bearing of 195, that means he's heading southeast. And since z is directly east of earth, we can draw another right-angled triangle with y, z, and a right angle at y.
In both triangles, we have a side of length 20km. Now, to find the distance of y from z, we want to find the length of the hypotenuse of the second triangle, which is the side opposite the right angle at y.
By using the Pythagorean theorem, we can find this distance. The formula is c^2 = a^2 + b^2, where c is the hypotenuse and a and b are the other two sides.
Since we already know that one side is 20km, we can call the other side of the triangle "d" (the distance of y from z). The equation becomes d^2 = 20^2 + 20^2.
Now, all we have to do is solve for d. Plugging in the values, we get d^2 = 400 + 400, which simplifies to d^2 = 800.
Taking the square root of both sides, we find that d is approximately equal to 28.3km (to 3 significant figures).
So, the distance of y from z is approximately 28.3km. And as a clown, I assure you that it's a straight (or slightly angled, but still straight) path from y to z!
To calculate the distance from point Y to point Z, we can use trigonometry and the given bearings.
First, let's draw a rough diagram to understand the situation. Here's how it looks:
Y
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/
/
X /
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Z
The bearing of 060 means that the traveler moves at an angle of 60 degrees clockwise from the reference direction (North). Similarly, the bearing of 195 means that the traveler moves at an angle of 15 degrees clockwise from the reference direction (North).
Next, let's calculate the distance between points X and Y. We can use the cosine rule, as we have a triangle with the side lengths of 20km (opposite to angle 60 degrees), the unknown side (between X and Y), and the angle of 180 - 60 = 120 degrees (since the sum of the interior angles of a triangle is 180 degrees).
Applying the cosine rule:
Distance between X and Y squared = (20)^2 + (20)^2 - 2 * 20 * 20 * cos(120)
Distance between X and Y = sqrt((20)^2 + (20)^2 - 2 * 20 * 20 * cos(120))
Now, let's calculate the distance between Y and Z. We can use the sine rule since we have a triangle with the side lengths of the distance between X and Y (as calculated above), the unknown side (between Y and Z), and the angle of 180 - 195 = -15 degrees (since the sum of the interior angles of a triangle is 180 degrees).
Applying the sine rule:
Distance between Y and Z / sin(195) = Distance between X and Y / sin(-15)
Distance between Y and Z = (Distance between X and Y * sin(195)) / sin(-15)
Finally, substitute the value of the distance between X and Y into the formula for distance between Y and Z:
Distance between Y and Z = (sqrt((20)^2 + (20)^2 - 2 * 20 * 20 * cos(120)) * sin(195)) / sin(-15)
Calculating this expression will give you the distance between Y and Z, rounded to 3 significant figures.
To calculate the distance of village y from village z, we need to find the distance traveled by the man on each bearing.
1. Village X to Village Y (bearing 060):
The man travels 20 km from village X to village Y on a bearing of 060.
2. Village Y to Village Z (bearing 195):
Since village Z is directly east of Earth, the bearing of 195 implies the man is traveling south (180 degrees) while slightly to the west (15 degrees), forming a right-angled triangle with the distance traveled on the previous leg (XY).
To find the distance of Y from Z, we can use trigonometry. The side adjacent to the 15-degree angle is the distance between Y and Z, and the hypotenuse is the distance from X to Y.
Using the cosine rule, we have:
cos(15 degrees) = adjacent/hypotenuse
Let's calculate the adjacent side (Y to Z):
adjacent = cos(15 degrees) * hypotenuse
The hypotenuse is the distance from X to Y, which is 20 km.
adjacent = cos(15 degrees) * 20 km
Calculating the adjacent side:
adjacent = 0.9659 * 20 km ≈ 19.318 km
Therefore, the distance of Y from Z is approximately 19.318 km.
To find the distance of Z from Y, we can subtract this value from the total distance traveled on the bearing of 195 degrees.
Subtracting the distance traveled on the YZ leg from the total distance:
Z from Y = 20 km - 19.318 km = 0.682 km
Therefore, the distance of Z from Y is approximately 0.682 km.