A man travels from x on a bearing of 060 to a town y, which is 200km away. At y,he changes course and goes to a town z in a bearing of 195.if z is directly east of x, calculate
I. How far is y from z
ii. The distance of z from x
no, y is on a bearing of 60° from x.
The man travels on a heading of 60°
Draw a diagram. In ∆XYZ, you have angles
X = 30°
Y = 45°
Z = 105°
so, using the law of sines,
y/sin30° = 200/sin105° = z/sin105°
X=25.67
a traveler move from a town p on a bearing of 055 to a town q 200km away .he then move from q on a bearing of 155 to a town 400km from q.find the distance between p and r
To calculate the distance between two points on a coordinate plane, we can use the distance formula, which is based on the Pythagorean theorem.
For this question, we can break it down into two parts:
I. Calculate how far y is from z.
II. Calculate the distance of z from x.
Now let's calculate each part step by step:
Part I: How far is y from z?
1. Start by drawing a diagram to visualize the problem.
- Place point x as the starting point.
- Mark town y on a bearing of 060 from x, 200 km away.
- Mark town z on a bearing of 195 from y.
2. Find the angle between the bearings of y and z.
- The angle between a bearing of 060 and a bearing of 195 is 195 - 060 = 135 degrees.
3. Use the Law of Cosines to determine the distance between y and z.
- The Law of Cosines states that c^2 = a^2 + b^2 - 2ab * cos(C), where c is the side opposite the angle C.
- Let's assume side c is the distance between y and z and side a is 200 km (from x to y).
- Rearrange the equation to solve for c: c^2 = a^2 + b^2 - 2ab * cos(C) => c^2 = 200^2 + 200^2 - 2 * 200 * 200 * cos(135).
- Calculate c^2 using trigonometric functions or a calculator.
4. Take the square root of c^2 to determine the actual distance between y and z.
Part II: What is the distance of z from x?
1. Since z is directly east of x, it means that the bearing from x to z is 090 degrees.
2. Use the Law of Sines to calculate the distance between x and z.
- The Law of Sines states that sin(A) / a = sin(B) / b = sin(C) / c, where A, B, and C are angles opposite respective sides a, b, and c.
- In this case, angle A is 30 degrees (180 - bearing of y) and side a is 200 km (from x to y).
- Angle B is 45 degrees (135 degrees / 2) and side b is the distance from y to z (from Part I).
- Rearrange the equation to solve for side c (distance from x to z): sin(C) = c * sin(A) / a.
- Substitute the known values into the equation: sin(45) = c * sin(30) / 200.
- Solve for side c using trigonometric functions or a calculator.
By following these steps, you should be able to calculate the distances between y and z, as well as between x and z.