Scorpion Gulch and Western Railway are preparing to build a new line through Rolling Mountains. They have hired you to do some calculations for tunnels and bridges needed on the line. You set up a Cartesian coordinate system with its origin at the entrance to the tunnel through Bald Mountain. Your surveying crew finds that the mountain rises 250 m above the level or the track and that the next valley goes down 50 m bellow the level of the track. The cross section of the mountain and the valley is roughly sinusoidal with a horizontal distance of 700 m from the top of the mountain to the bottom of the valley.
a. Write an equation expressing the vertical distance y from the track to the surface of the mountain or valley in terms of the horizontal distance x from the tunnel entrance.
b. How long will the tunnel be?
c. How long will the bridge be?
d. The company thinks it might be cheaper to build the line if the entire project is raised to y=20, thus making the tunnel shorter and the bridge longer. Find the new values of x at the ends of the tunnel and bridge. Then find the new lengths of each.
Honestly, this problem is such a joke that I don't even think Mahatma Ghandi could solve it.
a. To express the vertical distance y from the track to the surface of the mountain or valley in terms of the horizontal distance x from the tunnel entrance, we can use a sinusoidal function. Since the cross section of the mountain and the valley is roughly sinusoidal, the equation can be written as:
y = A * sin(B * x + C) + D
where A, B, C, and D are constants that determine the characteristics of the sinusoidal curve.
Given:
- The mountain rises 250 m above the level of the track, so D = 250.
- The next valley goes down 50 m below the level of the track, so D = -50.
- The horizontal distance from the top of the mountain to the bottom of the valley is 700 m, so when x = 0, we should be at the top of the mountain.
To find the values of A, B, and C, we can use the information given:
- The distance between the peak and the trough is the amplitude of the sinusoidal function, so A = (250 - (-50))/2 = 150.
- The period of the sinusoidal function is the distance from the top of the mountain to the bottom of the valley, so B = 2π/700.
- The phase shift of the sinusoidal function can be determined by finding the value of x at the peak (since C always represents the horizontal shift to the starting point of the sinusoidal function). At the peak, y = 250, so we can substitute these values into the equation to solve for C:
250 = 150 * sin(B * x + C) + 250
0 = 150 * sin(B * x + C)
Solving for C, we find C = -π/2.
Therefore, the equation expressing the vertical distance y is:
y = 150 * sin((2π/700) * x - π/2) + 250 (for the mountain)
y = -150 * sin((2π/700) * x - π/2) - 50 (for the valley)
b. The tunnel goes through the mountain until the track is level with the ground. So, to find the length of the tunnel, we need to find the values of x where y = 0.
150 * sin((2π/700) * x - π/2) + 250 = 0
Solving for x, we find:
sin((2π/700) * x - π/2) = -5/3
Since sin(-π/2) = -1 and sin(π/2) = 1, the above equation has no solutions within the given range for x. This means that the tunnel does not reach ground level, and the train line will always pass through the mountain.
c. Since the tunnel does not reach ground level, a bridge will be needed to cross the valley. The length of the bridge will be the horizontal distance across the valley, which is 700 m.
d. If the entire project is raised to y = 20, the equation for the vertical distance y will change. The new equation will be:
y_new = A * sin(B * x + C_new) + 20
The new values of x at the ends of the tunnel and bridge can be found by substituting y = 20 into the equation and solving for x.
20 = 150 * sin((2π/700) * x - π/2) + 20
Solving for x, we find:
sin((2π/700) * x - π/2) = 0
Since sin(0) = 0, the above equation has solutions when (2π/700) * x - π/2 = 0.
(2π/700) * x - π/2 = 0
(2π/700) * x = π/2
x_new = (700/2) = 350
Therefore, the new values of x at the ends of the tunnel and bridge are 0 and 350.
To find the new lengths of each, we calculate the length based on the difference between the x-values:
Length of the tunnel = 350 - 0 = 350 m
Length of the bridge = 700 - 350 = 350 m
a. To express the vertical distance y from the track to the surface of the mountain or valley in terms of the horizontal distance x from the tunnel entrance, we can use the equation of a sinusoidal function. Let's assume the top of the mountain is at x = 0.
The equation for a sinusoidal function in standard form is:
y = A*sin(Bx + C) + D
where:
A is the amplitude of the function
B is the horizontal scaling factor
C is the phase shift
D is the vertical shift
In this case, the vertical distance from the track to the surface of the mountain or valley changes between 250 m (mountain) and -50 m (valley). Therefore, the amplitude of the sinusoidal function is:
A = |250 - (-50)|/2 = 300/2 = 150
The period (horizontal distance for one full cycle) of the sinusoidal function is 700 m. The horizontal scaling factor (B) is determined by the formula:
B = (2*pi)/period = (2*pi)/700
Since the top of the mountain is at x = 0, C is 0. The vertical shift (D) is the average of the maximum and minimum y-values, which is (250 + (-50))/2 = 100.
Therefore, the equation expressing the vertical distance y in terms of the horizontal distance x is:
y = 150*sin((2*pi*x)/700) + 100
b. To find the length of the tunnel, we need to determine the range of x-values where the vertical distance is below the track level. This occurs when y is less than or equal to 0. So, we need to solve the equation:
150*sin((2*pi*x)/700) + 100 ≤ 0
Simplifying the inequality gives:
sin((2*pi*x)/700) ≤ -2/3
To solve this inequality, you can convert it to an equation and find the x-values where the sine function is equal to -2/3. Use inverse trigonometric functions (arcsin) to find the angles corresponding to this value, and then solve for x using the formula:
x = (700/(2*pi)) * arcsin(-2/3)
Once you find the values of x where the inequality is satisfied, the length of the tunnel will be the difference between the maximum and minimum values of x.
c. To find the length of the bridge, we need to determine the range of x-values where the vertical distance is above the track level. This occurs when y is greater than or equal to 0. So, we need to solve the equation:
150*sin((2*pi*x)/700) + 100 ≥ 0
Simplifying the equation gives:
sin((2*pi*x)/700) ≥ -2/3
Again, you can use inverse trigonometric functions (arcsin) to find the angles corresponding to -2/3, and then solve for x using the formula:
x = (700/(2*pi)) * arcsin(-2/3)
Once you find the values of x where the equation is satisfied, the length of the bridge will be the difference between the maximum and minimum values of x.
d. To find the new values of x at the ends of the tunnel and bridge when the project is raised to y = 20, we need to solve the equation:
150*sin((2*pi*x)/700) + 100 = 20
Subtracting 100 from both sides and solving for sin((2*pi*x)/700) gives:
sin((2*pi*x)/700) = -80/150
Using inverse trigonometric functions (arcsin), find the angles corresponding to -80/150, and then solve for x using the formula:
x = (700/(2*pi)) * arcsin(-80/150)
Once you find the values of x where the equation is satisfied, the new lengths of the tunnel and bridge will be the differences between the maximum and minimum values of x.