A tunnel for a new highway is to be cut through a mountain that is 260 feet high. At a distance of 200 feet from the base of the mountain, the angle of elevation is 36 degrees. From a distance of 150 feet on the other side of the mountain, the angle of elevation is 47 degrees. Approximate the length of the tunnel to the nearest foot.
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To approximate the length of the tunnel, we can use trigonometry.
Let's denote the length of the tunnel as "x" feet.
Using the angle of elevation of 36 degrees, we can form a right triangle with the height of the mountain (260 feet) and the distance from the base of the mountain to the tunnel entrance (200 feet).
In this triangle, the opposite side (height of the mountain) is known, and the adjacent side (distance from the base of the mountain to the tunnel entrance) is known. We can use the tangent function:
tan(36 degrees) = opposite/adjacent
Using the given values:
tan(36 degrees) = 260/200
Now, we can solve for the opposite side to find the length of the tunnel:
x + 150 = 200 / tan(36 degrees)
x + 150 = 200 / 0.7265
x + 150 = 275.3212
x = 275.3212 - 150
x = 125.3212
Therefore, the approximate length of the tunnel is 125 feet (rounded to the nearest foot).
To find the length of the tunnel, we can use the concept of trigonometry and the angles of elevation given.
Let's call the height of the mountain H and the length of the tunnel L.
From the given information, we can form two right-angled triangles: one on the side with the angle of elevation of 36 degrees and another on the side with the angle of elevation of 47 degrees.
For the triangle with a 36-degree angle of elevation:
The height of the mountain (H) is the opposite side and the distance from the base to the observer (200 feet) is the adjacent side. We need to find the length of the tunnel (L), which is the hypotenuse.
Using the tangent function:
tan(36) = H ÷ 200
H = 200 * tan(36)
Similarly, for the triangle with a 47-degree angle of elevation:
The height of the mountain (H) is the opposite side and the distance from the base to the observer (150 feet) is the adjacent side. Again, we need to find the length of the tunnel (L), which is the hypotenuse.
Using the tangent function:
tan(47) = H ÷ 150
H = 150 * tan(47)
Now we have two equations to find H. Once we find H, we can calculate the length of the tunnel (L) by adding the two distances, 200 feet and 150 feet.
To perform the calculations, we can use a scientific calculator or an online trigonometry calculator. Let's approximate the values to the nearest foot.
Using a calculator, we find:
H ≈ 260 feet
L ≈ 200 + 150 = 350 feet
Therefore, the approximate length of the tunnel is 350 feet.