a tunnel under a river is 196.8 ft below the surface at its west point. if the angle of depression of the runnel is 4.962 then how far apart on the surface are entrances to the tunnel? how long is tunnel

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To find the distance between the entrances to the tunnel and the length of the tunnel, we can use trigonometry. Let's assume the distance between the entrances is represented by "x" and the length of the tunnel is represented by "y".

Step 1: Determine the right-angled triangle
We have a right-angled triangle formed by the vertical distance (196.8 ft), the horizontal distance (x), and the angle of depression (4.962).

Step 2: Use trigonometric function
The tangent function can be used to determine the value of x. The formula for tangent is:

tan(angle) = opposite/adjacent

In this case, the opposite side is the vertical distance (196.8 ft), and the adjacent side is the horizontal distance (x).

Therefore, we have:

tan(4.962) = 196.8 / x

Step 3: Solve for x
Rearrange the equation to solve for x:

x = 196.8 / tan(4.962)

Using a calculator, we can find that tan(4.962) is approximately 0.086.

x = 196.8 / 0.086
x ≈ 2283.72 ft

So, the entrances to the tunnel are approximately 2283.72 ft apart on the surface.

Step 4: Determine the length of the tunnel
To find the length of the tunnel (y), we can use the tangent function again. The formula is the same:

tan(angle) = opposite/adjacent

In this case, the opposite side is the vertical distance (196.8 ft), and the adjacent side is the length of the tunnel (y).

Therefore, we have:

tan(4.962) = 196.8 / y

Step 5: Solve for y
Rearrange the equation to solve for y:

y = 196.8 / tan(4.962)

Using a calculator, tan(4.962) is approximately 0.086.

y = 196.8 / 0.086
y ≈ 2283.72 ft

So, the length of the tunnel is approximately 2283.72 ft.

To find the distance between the entrances of the tunnel on the surface, we can use trigonometry and the concept of angle of depression.

Let's first understand the situation. We have a tunnel that runs under a river. At its west point, the tunnel is 196.8 ft below the surface. The given angle of depression is 4.962 degrees.

To solve this problem, we'll use the tangent function, which relates the angle of depression to the distance and the height. The tangent of an angle is the ratio of the opposite side (height in this case) to the adjacent side (distance on the surface).

Let's assign variables:

x = distance between the entrances on the surface
h = height of the tunnel below the surface (196.8 ft)

Now, we can use the tangent function:
tan(angle) = opposite/adjacent

Using the given angle of depression, we have:
tan(4.962°) = h / x

We know h is 196.8 ft, so we have:
tan(4.962°) = 196.8 / x

We can solve for x by rearranging the equation:
x = 196.8 / tan(4.962°)

Using a calculator, we find:
x ≈ 2250.36 ft

Therefore, the distance between the entrances on the surface is approximately 2250.36 ft.

To find the length of the tunnel, we need to find the hypotenuse of the right triangle formed by the height of the tunnel and the distance between the entrances on the surface. We can use the Pythagorean theorem to find the length.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let's assign variables:

c = length of the tunnel (hypotenuse)
h = height of the tunnel below the surface (196.8 ft)
x = distance between the entrances on the surface (2250.36 ft)

Using the Pythagorean theorem, we have:
c^2 = h^2 + x^2

Substituting the values we found earlier:
c^2 = 196.8^2 + 2250.36^2

Calculating this expression, we find:
c ≈ 2251.66 ft

Therefore, the length of the tunnel is approximately 2251.66 ft.

Assume a V-shaped profile,with the low point in the middle of the tunnel. If the tunnel length is L,

tan 4.962 = 196.8 ft/(L/2)
0.08682 *L/2 = 196.8
Solve for L