A sinusoidal section is to be built as part of a roller coaster track. The high point is h=27 m at a horizontal distance d=0 m. It has a low point at h=-3 m at d=50 m.
a. Write an equation expressing h in terms of d.
b. The lengths of the horizontal timbers used to build the roller coaster's supporting structure are values of d. Calculate the length of the horizontal timber that is 4 m above the ground.
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a. To write an equation expressing h in terms of d, we can use the equation for a sinusoidal function with the following form:
h = A * sin(B * (d - C)) + D
where A is the amplitude, B is the period, C is the phase shift, and D is the vertical shift.
Given that the high point is h = 27 m at d = 0 m, we can substitute these values into the equation:
27 = A * sin(B * (0 - C)) + D
Since sin(0) = 0, the equation becomes:
27 = A * 0 + D
Therefore, D = 27.
Similarly, the low point is h = -3 m at d = 50 m:
-3 = A * sin(B * (50 - C)) + 27
Now we have two equations:
1) 27 = A * sin(B * (0 - C)) + 27
2) -3 = A * sin(B * (50 - C)) + 27
Simplifying these equations, we get:
1) 0 = A * sin(B * (0 - C))
2) -30 = A * sin(B * (50 - C))
We see that sin(B * (0 - C)) must be zero, as it is equal to 0/A. This means that B * (0 - C) must be an integer multiple of π (pi).
To calculate the period, we can use the fact that the distance between the high point and the low point is equal to one full wavelength, which is the period. In this case, the distance is 50 m, so the period is 50 m.
Therefore, B * (50 - C) = k * π, where k is an integer.
Now we have two equations:
1) B * (0 - C) = n * π, where n is an integer.
2) B * (50 - C) = k * π, where k is an integer.
By subtracting equation 1 from equation 2, we can eliminate B:
B * (50 - C) - B * (0 - C) = k * π - n * π
We simplify:
50B - BC = (k - n) * π
To find the values of A and B, we need more information about the graph. Once we have values for A and B, we can substitute them into the equation:
h = A * sin(B * (d - C)) + 27
b. To calculate the length of the horizontal timber that is 4 m above the ground, we can plug the given value of h = 4 m into the equation and solve for d.