IDNLearn.com is designed to help you find accurate answers with ease. Ask anything and receive prompt, well-informed answers from our community of knowledgeable experts.
Sagot :
Let's tackle the problem presented by the surveyor step-by-step.
## Part (a):
We start with point [tex]\(Y\)[/tex] being equidistant from points [tex]\(X\)[/tex] and [tex]\(Z\)[/tex], therefore [tex]\(XY = YZ = x\)[/tex].
The problem requires us to find [tex]\(XZ\)[/tex] in terms of [tex]\(x\)[/tex] and [tex]\(\theta\)[/tex].
To solve this, we will use the Law of Cosines, which states:
[tex]\[ c^2 = a^2 + b^2 - 2ab\cos(\theta) \][/tex]
In this scenario:
- [tex]\(a = XY = x\)[/tex]
- [tex]\(b = YZ = x\)[/tex]
- [tex]\(\theta = \angle XYZ\)[/tex]
- [tex]\(c = XZ\)[/tex]
Plugging these values into the Law of Cosines, we get:
[tex]\[ XZ^2 = XY^2 + YZ^2 - 2 \cdot XY \cdot YZ \cdot \cos(\theta) \][/tex]
Substituting [tex]\(XY = x\)[/tex] and [tex]\(YZ = x\)[/tex]:
[tex]\[ XZ^2 = x^2 + x^2 - 2 \cdot x \cdot x \cdot \cos(\theta) \][/tex]
Simplify the equation:
[tex]\[ XZ^2 = 2x^2 - 2x^2 \cos(\theta) \][/tex]
[tex]\[ XZ^2 = 2x^2 (1 - \cos(\theta)) \][/tex]
Taking the square root on both sides to solve for [tex]\(XZ\)[/tex]:
[tex]\[ XZ = \sqrt{2x^2(1 - \cos(\theta))} \][/tex]
[tex]\[ XZ = x \sqrt{2(1 - \cos(\theta))} \][/tex]
Hence, we have shown that:
[tex]\[ XZ = x \sqrt{2(1 - \cos(\theta))} \][/tex]
## Part (b):
Now, we need to calculate [tex]\(XZ\)[/tex] if [tex]\(x = 240 \, \text{km}\)[/tex] and [tex]\(\theta = 132^\circ\)[/tex].
### Step-by-step Calculation:
1. Convert [tex]\(\theta\)[/tex] from degrees to radians:
[tex]\[ \theta = 132^\circ \][/tex]
The conversion formula is [tex]\( \theta \text{ radians} = \theta \text{ degrees} \times \left(\frac{\pi}{180}\right) \)[/tex]:
[tex]\[ \theta = 132 \times \left(\frac{\pi}{180}\right) \approx 2.303 \text{ radians} \][/tex]
2. Calculate [tex]\(\cos(\theta)\)[/tex]:
[tex]\[ \cos(132^\circ) = \cos(2.303 \text{ radians}) \approx -0.669 \][/tex]
3. Plug these values into the formula [tex]\(XZ = x \sqrt{2(1 - \cos(\theta))}\)[/tex]:
[tex]\[ XZ = 240 \, \text{km} \times \sqrt{2(1 - (-0.669))} \][/tex]
[tex]\[ XZ = 240 \times \sqrt{2(1 + 0.669)} \][/tex]
[tex]\[ XZ = 240 \times \sqrt{2(1.669)} \][/tex]
[tex]\[ XZ = 240 \times \sqrt{3.338} \][/tex]
[tex]\[ XZ \approx 240 \times 1.827 \][/tex]
[tex]\[ XZ \approx 438.48 \][/tex]
4. Rounding to the nearest kilometer:
[tex]\[ XZ \approx 439 \, \text{km} \][/tex]
Therefore, the distance between points [tex]\(X\)[/tex] and [tex]\(Z\)[/tex] is approximately [tex]\(439\)[/tex] kilometers.
## Part (a):
We start with point [tex]\(Y\)[/tex] being equidistant from points [tex]\(X\)[/tex] and [tex]\(Z\)[/tex], therefore [tex]\(XY = YZ = x\)[/tex].
The problem requires us to find [tex]\(XZ\)[/tex] in terms of [tex]\(x\)[/tex] and [tex]\(\theta\)[/tex].
To solve this, we will use the Law of Cosines, which states:
[tex]\[ c^2 = a^2 + b^2 - 2ab\cos(\theta) \][/tex]
In this scenario:
- [tex]\(a = XY = x\)[/tex]
- [tex]\(b = YZ = x\)[/tex]
- [tex]\(\theta = \angle XYZ\)[/tex]
- [tex]\(c = XZ\)[/tex]
Plugging these values into the Law of Cosines, we get:
[tex]\[ XZ^2 = XY^2 + YZ^2 - 2 \cdot XY \cdot YZ \cdot \cos(\theta) \][/tex]
Substituting [tex]\(XY = x\)[/tex] and [tex]\(YZ = x\)[/tex]:
[tex]\[ XZ^2 = x^2 + x^2 - 2 \cdot x \cdot x \cdot \cos(\theta) \][/tex]
Simplify the equation:
[tex]\[ XZ^2 = 2x^2 - 2x^2 \cos(\theta) \][/tex]
[tex]\[ XZ^2 = 2x^2 (1 - \cos(\theta)) \][/tex]
Taking the square root on both sides to solve for [tex]\(XZ\)[/tex]:
[tex]\[ XZ = \sqrt{2x^2(1 - \cos(\theta))} \][/tex]
[tex]\[ XZ = x \sqrt{2(1 - \cos(\theta))} \][/tex]
Hence, we have shown that:
[tex]\[ XZ = x \sqrt{2(1 - \cos(\theta))} \][/tex]
## Part (b):
Now, we need to calculate [tex]\(XZ\)[/tex] if [tex]\(x = 240 \, \text{km}\)[/tex] and [tex]\(\theta = 132^\circ\)[/tex].
### Step-by-step Calculation:
1. Convert [tex]\(\theta\)[/tex] from degrees to radians:
[tex]\[ \theta = 132^\circ \][/tex]
The conversion formula is [tex]\( \theta \text{ radians} = \theta \text{ degrees} \times \left(\frac{\pi}{180}\right) \)[/tex]:
[tex]\[ \theta = 132 \times \left(\frac{\pi}{180}\right) \approx 2.303 \text{ radians} \][/tex]
2. Calculate [tex]\(\cos(\theta)\)[/tex]:
[tex]\[ \cos(132^\circ) = \cos(2.303 \text{ radians}) \approx -0.669 \][/tex]
3. Plug these values into the formula [tex]\(XZ = x \sqrt{2(1 - \cos(\theta))}\)[/tex]:
[tex]\[ XZ = 240 \, \text{km} \times \sqrt{2(1 - (-0.669))} \][/tex]
[tex]\[ XZ = 240 \times \sqrt{2(1 + 0.669)} \][/tex]
[tex]\[ XZ = 240 \times \sqrt{2(1.669)} \][/tex]
[tex]\[ XZ = 240 \times \sqrt{3.338} \][/tex]
[tex]\[ XZ \approx 240 \times 1.827 \][/tex]
[tex]\[ XZ \approx 438.48 \][/tex]
4. Rounding to the nearest kilometer:
[tex]\[ XZ \approx 439 \, \text{km} \][/tex]
Therefore, the distance between points [tex]\(X\)[/tex] and [tex]\(Z\)[/tex] is approximately [tex]\(439\)[/tex] kilometers.
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Your questions deserve accurate answers. Thank you for visiting IDNLearn.com, and see you again for more solutions.
