Question

Using trigonometry to find a length in a word problem with two right triangles

Answer 1

Given the word problem, we can deduce the following information:

1. The surveyor is located 340 meters from one entrance of the tunnel at an angle of 58°.

2. He is 193 meters from the other entrance of the tunnel, at an angle of 21°.

To determine the length of the tunnel, we find the length of the remaining sides of the triangles first. Based on the given figure, we can form two triangles:

Triangle 1:

We let x be the opposite side of the angle (58°). To find x, we use the formula:

We plug in what we know:

`\begin{gathered} \sin 58\degree=(x)/(340) \\ \text{Simplify and rearrange} \\ x=340(\sin 58\degree) \\ x=288.336\text{ meters} \end{gathered}`

For Triangle 2:

We apply the same process. So,

`\begin{gathered} \sin 21\degree=(y)/(193) \\ \text{Simplify and rearrange} \\ y=193(\sin 21\degree) \\ y=\text{ 69.165 meters} \end{gathered}`

Next, we get the total length using the process below:

Total = x+y =288.336+69.165=357.5 meters

Therefore, the answer is 357.5 meters.