Answered

Discover how IDNLearn.com can help you find the answers you need quickly and easily. Join our community to receive prompt and reliable responses to your questions from experienced professionals.

5. A surveyor is trying to determine the distance between points [tex]\( X \)[/tex] and [tex]\( Z \)[/tex]. However, the distance cannot be determined directly as a ridge lies between the two points. From a point [tex]\( Y \)[/tex], which is equidistant from [tex]\( X \)[/tex] and [tex]\( Z \)[/tex], he measures the angle [tex]\( XY Z \)[/tex].

a) If [tex]\( XY = x \)[/tex] and [tex]\( \angle XYZ = \theta \)[/tex], show that [tex]\( XZ = x \sqrt{2(1 - \cos \theta)} \)[/tex].

b) Calculate [tex]\( XZ \)[/tex] (to the nearest kilometer) if [tex]\( x = 240 \text{ km} \)[/tex] and [tex]\( \theta = 132^{\circ} \)[/tex].

Sagot :

GinnyAnsweridn
Certainly! Let's break down this problem step by step.

### Part (a)

We need to show that [tex]\( XZ = x \sqrt{2(1 - \cos \theta)} \)[/tex].

Given:
- [tex]\(XY = x\)[/tex]
- [tex]\(\angle XYZ = \theta\)[/tex]

Since we have points [tex]\(X\)[/tex], [tex]\(Y\)[/tex], and [tex]\(Z\)[/tex], with [tex]\(Y\)[/tex] equidistant from [tex]\(X\)[/tex] and [tex]\(Z\)[/tex], we can use the Law of Cosines to find the distance [tex]\(XZ\)[/tex].

The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(\gamma) \][/tex]

For our triangle [tex]\(XYZ\)[/tex]:
- [tex]\(a = b = XY = x\)[/tex]
- [tex]\(\gamma = \angle XYZ = \theta\)[/tex]
- [tex]\(c = XZ\)[/tex]

Applying these to the Law of Cosines:
[tex]\[ XZ^2 = XY^2 + YZ^2 - 2 \cdot XY \cdot YZ \cdot \cos(\theta) \][/tex]

Since [tex]\(XY = YZ = x\)[/tex]:
[tex]\[ XZ^2 = x^2 + x^2 - 2 \cdot x \cdot x \cdot \cos(\theta) \][/tex]
[tex]\[ XZ^2 = 2x^2 - 2x^2 \cdot \cos(\theta) \][/tex]
[tex]\[ XZ^2 = 2x^2 (1 - \cos(\theta)) \][/tex]

Taking the square root of both sides:
[tex]\[ XZ = \sqrt{2x^2 (1 - \cos(\theta))} \][/tex]
[tex]\[ XZ = x \sqrt{2(1 - \cos(\theta))} \][/tex]

Thus, we have shown that:
[tex]\[ XZ = x \sqrt{2(1 - \cos(\theta))} \][/tex]

### Part (b)

Now, we need to calculate [tex]\(XZ\)[/tex] to the nearest kilometre given that:
- [tex]\(x = 240 \, \text{km}\)[/tex]
- [tex]\(\theta = 132^\circ\)[/tex]

Using the formula derived in part (a):
[tex]\[ XZ = x \sqrt{2(1 - \cos(\theta))} \][/tex]

First, convert [tex]\(\theta\)[/tex] from degrees to radians because trigonometric functions typically use radians.

[tex]\[ \theta = 132^\circ \][/tex]
[tex]\[ \theta_{\text{radians}} = \frac{132 \times \pi}{180} \approx 2.303 \, \text{radians} \][/tex]

Now, compute [tex]\(\cos(132^\circ)\)[/tex]:
[tex]\[ \cos(132^\circ) \approx -0.6691 \][/tex]

Substitute the values into the formula:
[tex]\[ XZ = 240 \sqrt{2(1 - (-0.6691))} \][/tex]
[tex]\[ XZ = 240 \sqrt{2(1 + 0.6691)} \][/tex]
[tex]\[ XZ = 240 \sqrt{2(1.6691)} \][/tex]
[tex]\[ XZ = 240 \sqrt{3.3382} \][/tex]
[tex]\[ XZ \approx 240 \times 1.832 \][/tex]
[tex]\[ XZ \approx 439 \, \text{km} \][/tex]

Hence, the distance [tex]\(XZ\)[/tex] is approximately 439 kilometres to the nearest kilometre.
Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. IDNLearn.com has the answers you need. Thank you for visiting, and we look forward to helping you again soon.