How high is the peak of the Matterhorn? Below is a picture of the Matterhorn in Switzerland. As a surveyor, it is your job to estimate the elevation of the mountain. Using a device called a theodolite, you measure two angles from the horizontal plane to the top of the mountain as shown below. You also record the distance between the two measurements to be 1500 ft. Your elevation is known to be at 9,756ft above sea level. Determine the height of the Matterhorn (above sea level) to the nearest foot.

Answer:

[tex]E\approx14,962\text{ feet}[/tex]

Step-by-step explanation:

Please refer to the attachment below.

As shown, we can split the diagram into a scalene triangle and a right triangle.

The strategy here is to use the Law of Sines to find x, then use right triangle ratios to find h, the height of Matterhorn (with respect to our elevation).

Let’s start with the scalene triangle.

To use the Law of Sines, we will need two angles and the sides opposite to the angles.

Since we need to find x, we need the angle opposite to x. This is already given to us as 30°.

We can use the side that measures 1,500. The angle opposite to it θ is unknown.

So, we will need to determine θ. From supplementary angles, we know that:

[tex]\alpha+35=180[/tex]

So, it follows that:

[tex]\alpha=145^\circ[/tex]

The interior angles of a triangle always sum to 180°. Hence:

[tex]30+\alpha+\theta=180[/tex]

By substitution:

[tex]30+145+\theta=180[/tex]

Hence:

[tex]\theta=5^\circ[/tex]

Now, we can use the Law of Sines, given by:

[tex]\displaystyle \frac{\sin(A)}{a}=\frac{\sin(B)}{b}[/tex]

It is important to align the angles correctly with the sides.

So, we will substitute θ = 5° for A and 1,500 for a.

And we will substitute 30° for B and x for b. Hence:

[tex]\displaystyle \frac{\sin(5^\circ)}{1,500}=\frac{\sin(30^\circ)}{x}[/tex]

Solve for x. Cross-multiply:

[tex]x\sin(5^\circ)=1500\sin(30^\circ)[/tex]

Hence:

[tex]\displaystyle x=\frac{1500\sin(30^\circ)}{\sin(5^\circ)}[/tex]

When doing trigonometry (or any math, really), try not to round intermediate values, as this may result in inaccuracies in the final answer. Hence, we will leave this as is.

Now, we can use the right triangle.

We know that an angle is 35°.

The height h is opposite to our angle.

And x was our hypotenuse. Therefore, we can use the sine ratio given by:

[tex]\displaystyle \sin(x^\circ)=\frac{\text{Opposite}}{\text{Hypotenuse}}[/tex]

In this case, our angle is 35°.

The opposite side is h.

And the hypotenuse is x. Hence, by substitution:

[tex]\displaystyle \sin(35^\circ)=\frac{h}{x}[/tex]

Solve for h. Multiply both sides by x:

[tex]h=x\sin(35^\circ)[/tex]

Since we already know the value of x:

[tex]\displaystyle h=\frac{1500\sin(30^\circ)}{\sin(5^\circ)}\cdot\sin(35^\circ)[/tex]

Simplify. So, the height of Matterhorn with respect to our current elevation is:

[tex]\displaystyle h =\frac{1500\sin(30^\circ)\sin(35^\circ)}{\sin(5^\circ)}[/tex]

The total elevation of Matterhorn will be the current elevation of the surveyor plus the height with respect to the current elevation or simply h. So, the elevation height is:

[tex]E=9756+h[/tex]

By substitution:

[tex]\displaystyle E=9756+\frac{1500\sin(30^\circ)\sin(35^\circ)}{\sin(5^\circ)}[/tex]

Now we may approximate. Use a calculator. So, the total elevation of Matterhorn is:

[tex]E=14961.78867...\approx14,962\text{ feet}[/tex]

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