Three cities X,Y, and Z are connected by straight highways. City X is 6 km from city Y, city Y is 4 km from city Z, and city X is 5 km from city Z. Find the angle made by the highways XY and YZ?
Three cities X,Y, and Z are connected by straight highways. City X is 6 km from city Y, city Y is 4 km from city Z, and city X is 5 km from city Z. Find the angle made by the highways XY and YZ?
physics
To find the angle made by the highways XY and YZ, we can use the Law of Cosines.
Let's label the sides of the triangle as follows:
- Side XY, which is 6 km, as side a.
- Side YZ, which is 4 km, as side b.
- Side XZ, which is 5 km, as side c.
The Law of Cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C),
where c represents the side opposite to angle C.
In this case, we want to find angle C, which is the angle between highways XY and YZ.
Plugging in the given values, we have:
5^2 = 6^2 + 4^2 - 2 * 6 * 4 * cos(C).
Simplifying:
25 = 36 + 16 - 48 * cos(C).
Now, let's isolate cos(C):
48 * cos(C) = 36 + 16 - 25.
48 * cos(C) = 27.
cos(C) = 27/48.
cos(C) ≈ 0.5625.
To find angle C, we can take the inverse cosine (cos⁻¹) of 0.5625:
C ≈ cos⁻¹(0.5625).
Using a calculator, we find:
C ≈ 55.27 degrees.
Therefore, the angle made by the highways XY and YZ is approximately 55.27 degrees.
To find the angle made by highways XY and YZ, we need to use the concept of the Law of Cosines from trigonometry.
Here's how we can solve it step by step:
1. Label the three cities on a diagram with X, Y, and Z.
2. Note the distances given: X is 6 km from Y, Y is 4 km from Z, and X is 5 km from Z.
3. Now, let's focus on triangle XYZ. The sides are XY, YZ, and XZ.
4. According to the Law of Cosines, the square of the length of one side of a triangle is equal to the sum of the squares of the other two sides minus twice their product, multiplied by the cosine of the angle between them.
In this case, we want to find the angle at city Y, so we will use the Law of Cosines to find the angle opposite side XY.
5. Apply the Law of Cosines to triangle XYZ:
XZ^2 = XY^2 + YZ^2 - 2 * XY * YZ * cos(angle Y)
Substitute the given values:
5^2 = 6^2 + 4^2 - 2 * 6 * 4 * cos(angle Y)
6. Simplify the equation:
25 = 36 + 16 - 48 * cos(angle Y)
7. Rearrange the equation to isolate the cosine term:
-27 = -48 * cos(angle Y)
8. Solve for cos(angle Y):
cos(angle Y) = -27 / -48 = 9 / 16
9. Finally, find the angle Y using the inverse cosine function (cos^(-1)):
angle Y = cos^(-1)(9 / 16)
After evaluating this expression, you'll find the angle made by the highways XY and YZ.