A roof in the shape of an upside down cosine wave is to be buile to cover an arena. The arena is 24m wide, the height of the roof at either side wall is 5m, and the maximum height of the roof is 9m.
A) write a function that gives the height of the roof in terms of your distance from the left wall.
B) what is the height if the roof 10m from the left wall?
Please help....
The description is vague "cosine wave"
at x=-12, y=5, at x=0, y=9, at x=12, y=5
y= 5 + 4 cos((x-12)/48)
check that.
A) To write a function that gives the height of the roof in terms of the distance from the left wall, you can use the cosine function to model the shape of the roof. The general equation for a cosine wave is:
y = A * cos(B * x + C) + D,
where:
- A represents the amplitude of the wave (half the difference between the minimum and maximum values),
- B represents the frequency of the wave (controls how quickly the wave repeats),
- C represents a phase shift (can be used to shift the wave horizontally),
- D represents the vertical shift (can be used to move the wave up or down).
In this case, the amplitude (A) is calculated as half the difference between the maximum height (9m) and the side wall height (5m), which is (9 - 5) / 2 = 2m.
The frequency (B) can be determined by considering the width of the arena, which is 24m. Since the roof starts at the side wall, travels up to its maximum height at the center, and then comes down to the other side wall, the distance traveled in each cycle is 24m. Therefore, the frequency can be calculated as 2π divided by the width:
B = 2π / 24 = π / 12.
The phase shift (C) can be set to 0 in this case because the roof starts at the left wall.
The vertical shift (D) is equal to the side wall height (5m).
Combining all these values, the height of the roof as a function of the distance from the left wall (x) can be expressed as:
height(x) = 2 * cos((π / 12) * x) + 5.
B) To calculate the height of the roof 10m from the left wall, you can substitute the value of x = 10 into the equation:
height(10) = 2 * cos((π / 12) * 10) + 5.
Now, evaluating this expression:
height(10) = 2 * cos((π / 12) * 10) + 5
= 2 * cos(π / 3) + 5
= 2 * (-0.5) + 5
= -1 + 5
= 4.
Therefore, the height of the roof 10m from the left wall is 4m.