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:
[tex]\begin{gathered} \sin 58\degree=\frac{x}{340} \\ \text{Simplify and rearrange} \\ x=340(\sin 58\degree) \\ x=288.336\text{ meters} \end{gathered}[/tex]
For Triangle 2:
We apply the same process. So,
[tex]\begin{gathered} \sin 21\degree=\frac{y}{193} \\ \text{Simplify and rearrange} \\ y=193(\sin 21\degree) \\ y=\text{ 69.165 meters} \end{gathered}[/tex]
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.
