Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To solve this problem, we need to determine two key distances:
(a) The distance from [tex]\(X\)[/tex] to [tex]\(Z\)[/tex] (which we will call [tex]\(XZ\)[/tex]).
(b) The distance from [tex]\(Y\)[/tex] to [tex]\(Z\)[/tex] (which we will call [tex]\(YZ\)[/tex]).
Let's start solving step-by-step:
### Step 1: Understanding Bearings and Coordinates
1. Bearing from [tex]\(X\)[/tex] to [tex]\(Y\)[/tex]:
- Bearing [tex]\(N 45^\circ E\)[/tex] means the direction is [tex]\(45^\circ\)[/tex] from the north towards the east.
- This implies [tex]\(XY\)[/tex] forms an angle of [tex]\(45^\circ\)[/tex] with the positive x-axis (east direction).
2. Bearing from [tex]\(Y\)[/tex] to [tex]\(Z\)[/tex]:
- Bearing [tex]\(S 60^\circ E\)[/tex] means the direction is [tex]\(60^\circ\)[/tex] from the south towards the east.
- If we convert this bearing for trigonometric purposes, it can be seen as [tex]\(30^\circ\)[/tex] above the negative y-axis (south direction).
### Step 2: Convert Bearings to Cartesian Coordinates
Let’s define our coordinates:
- Let [tex]\(X\)[/tex] be the origin point: [tex]\( (0, 0) \)[/tex]
- Let [tex]\(Y\)[/tex] be point [tex]\(Y(x, y)\)[/tex]
Using the [tex]\(45^\circ\)[/tex] for [tex]\(XY:\)[/tex]
- [tex]\(Y_x = 200 \cos(45^\circ) = 200 \cdot \frac{\sqrt{2}}{2} = 100\sqrt{2}\)[/tex]
- [tex]\(Y_y = 200 \sin(45^\circ) = 200 \cdot \frac{\sqrt{2}}{2} = 100\sqrt{2}\)[/tex]
So, coordinates of [tex]\(Y\)[/tex] are approximately [tex]\( (141.42, 141.42) \)[/tex].
### Step 3: Calculating [tex]\(XZ\)[/tex]
Since [tex]\(Z\)[/tex] is directly east of [tex]\(X\)[/tex], it lies on the x-axis:
- We know that [tex]\(Z\)[/tex]’s x-coordinate must be determined by considering the horizontal distance traversed.
To determine this, consider the bearings and form an understanding:
- Distance from [tex]\(Y\)[/tex] to the east component due to [tex]\(S 60^\circ E\)[/tex] will be along the x-axis from [tex]\(Y\)[/tex].
Utilizing trigonometric relationships:
- [tex]\(Y_y / \tan(60^\circ)\)[/tex] determines the x-direction displacement from [tex]\(Y\)[/tex]
- So, [tex]\(Z_x from Y = 100\sqrt{2} / \sqrt{3} = \frac{100\sqrt{2}}{\sqrt{3}} \approx 81.65\)[/tex]
Therefore:
[tex]\[ Z_x = Y_x + Z_x from Y = 100\sqrt{2} + \frac{100\sqrt{2}}{\sqrt{3}} = 223.07 \][/tex]
Hence, [tex]\(XZ\)[/tex] distance is approximately [tex]\( 223.071 \)[/tex] km.
### Step 4: Calculating [tex]\(YZ\)[/tex]
Using sine for [tex]\(S 60^\circ E\)[/tex]:
- [tex]\(YZ\)[/tex] is calculated by understanding [tex]\( Y_y\ / \sin(60^\circ)\)[/tex]
[tex]\[ Y_y / \sin(60^\circ) = \( 100\sqrt{2} / (\sqrt(3)/2) = 163.299 \][/tex]
Therefore:
[tex]\[ YZ = 163.299 \ km \][/tex]
### Final Results
(a) The distance from [tex]\(X\)[/tex] to [tex]\(Z\)[/tex] is approximately [tex]\(223.071\)[/tex] km.
(b) The distance from [tex]\(Y\)[/tex] to [tex]\(Z\)[/tex] is approximately [tex]\(163.299\)[/tex] km.
(a) The distance from [tex]\(X\)[/tex] to [tex]\(Z\)[/tex] (which we will call [tex]\(XZ\)[/tex]).
(b) The distance from [tex]\(Y\)[/tex] to [tex]\(Z\)[/tex] (which we will call [tex]\(YZ\)[/tex]).
Let's start solving step-by-step:
### Step 1: Understanding Bearings and Coordinates
1. Bearing from [tex]\(X\)[/tex] to [tex]\(Y\)[/tex]:
- Bearing [tex]\(N 45^\circ E\)[/tex] means the direction is [tex]\(45^\circ\)[/tex] from the north towards the east.
- This implies [tex]\(XY\)[/tex] forms an angle of [tex]\(45^\circ\)[/tex] with the positive x-axis (east direction).
2. Bearing from [tex]\(Y\)[/tex] to [tex]\(Z\)[/tex]:
- Bearing [tex]\(S 60^\circ E\)[/tex] means the direction is [tex]\(60^\circ\)[/tex] from the south towards the east.
- If we convert this bearing for trigonometric purposes, it can be seen as [tex]\(30^\circ\)[/tex] above the negative y-axis (south direction).
### Step 2: Convert Bearings to Cartesian Coordinates
Let’s define our coordinates:
- Let [tex]\(X\)[/tex] be the origin point: [tex]\( (0, 0) \)[/tex]
- Let [tex]\(Y\)[/tex] be point [tex]\(Y(x, y)\)[/tex]
Using the [tex]\(45^\circ\)[/tex] for [tex]\(XY:\)[/tex]
- [tex]\(Y_x = 200 \cos(45^\circ) = 200 \cdot \frac{\sqrt{2}}{2} = 100\sqrt{2}\)[/tex]
- [tex]\(Y_y = 200 \sin(45^\circ) = 200 \cdot \frac{\sqrt{2}}{2} = 100\sqrt{2}\)[/tex]
So, coordinates of [tex]\(Y\)[/tex] are approximately [tex]\( (141.42, 141.42) \)[/tex].
### Step 3: Calculating [tex]\(XZ\)[/tex]
Since [tex]\(Z\)[/tex] is directly east of [tex]\(X\)[/tex], it lies on the x-axis:
- We know that [tex]\(Z\)[/tex]’s x-coordinate must be determined by considering the horizontal distance traversed.
To determine this, consider the bearings and form an understanding:
- Distance from [tex]\(Y\)[/tex] to the east component due to [tex]\(S 60^\circ E\)[/tex] will be along the x-axis from [tex]\(Y\)[/tex].
Utilizing trigonometric relationships:
- [tex]\(Y_y / \tan(60^\circ)\)[/tex] determines the x-direction displacement from [tex]\(Y\)[/tex]
- So, [tex]\(Z_x from Y = 100\sqrt{2} / \sqrt{3} = \frac{100\sqrt{2}}{\sqrt{3}} \approx 81.65\)[/tex]
Therefore:
[tex]\[ Z_x = Y_x + Z_x from Y = 100\sqrt{2} + \frac{100\sqrt{2}}{\sqrt{3}} = 223.07 \][/tex]
Hence, [tex]\(XZ\)[/tex] distance is approximately [tex]\( 223.071 \)[/tex] km.
### Step 4: Calculating [tex]\(YZ\)[/tex]
Using sine for [tex]\(S 60^\circ E\)[/tex]:
- [tex]\(YZ\)[/tex] is calculated by understanding [tex]\( Y_y\ / \sin(60^\circ)\)[/tex]
[tex]\[ Y_y / \sin(60^\circ) = \( 100\sqrt{2} / (\sqrt(3)/2) = 163.299 \][/tex]
Therefore:
[tex]\[ YZ = 163.299 \ km \][/tex]
### Final Results
(a) The distance from [tex]\(X\)[/tex] to [tex]\(Z\)[/tex] is approximately [tex]\(223.071\)[/tex] km.
(b) The distance from [tex]\(Y\)[/tex] to [tex]\(Z\)[/tex] is approximately [tex]\(163.299\)[/tex] km.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
