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.
I drew triangle ABC where BC is a horizontal (the road) and angle B is 36° , and is on the left side of the diagram.
Angle C = 47° , on the right side
D is on BC and is the altitude of the mountain of 260 feet
I let BD = 200+x , and DC = 150+y
in the left triangle,
tan 36° = 260/(200+x)
200+x = 260/tan 36
x = 260/tan36 - 200 = 157.86 feet
find y in the same way.....
length of tunnel = x+y
= ....
y=92.5
x=157.9
x+y=250.4ft
To approximate the length of the tunnel, we can use trigonometry and the concept of similar triangles.
Let's denote the length of the tunnel as L.
Using the given information, we can create two right triangles. The first triangle is formed by the mountain, the base, and the line of sight from the base to the top of the mountain. The second triangle is formed by the tunnel, the distance of 200 feet, and the line of sight from the other side of the mountain to the top.
Let's start by finding the height of the mountain.
In the first triangle, the angle of elevation is 36 degrees, and the distance from the base to the mountain is 200 feet. We can use the tangent function to find the height:
tan(36 degrees) = height / 200 feet
height = tan(36 degrees) * 200 feet ≈ 212.13 feet
Now, let's find the height from the other side of the mountain.
In the second triangle, the angle of elevation is 47 degrees. The distance from the other side of the mountain to the tunnel is 150 feet. Using the same logic as before, we can find the height:
tan(47 degrees) = height / 150 feet
height = tan(47 degrees) * 150 feet ≈ 175.31 feet
Since the two heights form a right triangle, their sum is equal to the length of the tunnel:
L = 212.13 feet + 175.31 feet ≈ 387.44 feet
Therefore, the approximate length of the tunnel is 387 feet when rounded to the nearest foot.
To approximate the length of the tunnel, we can use trigonometry and create a right triangle based on the given angles of elevation.
Let's start by drawing a diagram:
47°
/ |
/ | 150 ft
/ |
/__________|
/ |
260 ft / |
/_________________|
36° 200 ft
In the diagram, the tunnel represents the base of the triangle, and the mountain represents the hypotenuse. We need to determine the length of the tunnel.
Let's focus on the right triangle on the side where the angle of elevation is 36 degrees. We can use the tangent function to relate the angle of elevation to the lengths of the sides:
Tan(36°) = height of the tunnel / distance from the base of the mountain
So, tan(36°) = height of the tunnel / 200 ft
To find the height of the tunnel, we rearrange the formula:
Height of the tunnel = tan(36°) * 200 ft
Now, let's focus on the right triangle on the side where the angle of elevation is 47 degrees. Using the same process, we can determine the length of the tunnel.
Tan(47°) = height of the tunnel / distance from the other side of the mountain
So, tan(47°) = height of the tunnel / 150 ft
Rearranging the formula, we find:
Height of the tunnel = tan(47°) * 150 ft
Now that we have the height of the tunnel from both sides of the mountain, we can approximate the overall length of the tunnel by adding the two heights together:
Approximate length of the tunnel = (height from the first side) + (height from the other side)
Approximate length of the tunnel = (tan(36°) * 200 ft) + (tan(47°) * 150 ft)
Calculating this expression will give us the approximate length of the tunnel.