Precalculus Trigonometric Functions Name
Precalculus Trigonometric Functions Name
Precalculus Trigonometric Functions Name
You have been tasked with analyzing and improving the operation of an old Ferris wheel at Calcu-Now’s Amusement Park. The goal is to determine the current speed of the Ferris wheel, suggest necessary adjustments to meet safety standards, and design a system for tracking rider height and timing camera shots without conflicting with water spray features. Using mathematical reasoning, you will provide the necessary calculations and recommendations to enhance both safety and thrill factor.
Paper For Above instruction
Introduction
The overhaul of the Ferris wheel involves multiple aspects, including its speed, rider safety, camera timing, and water spray operations. Utilizing trigonometric functions and physics principles, we analyze the current status and propose improvements to create an exhilarating yet safe experience for visitors.
Determining Current Speed of the Ferris Wheel
The Ferris wheel's rotation time is given as 1 minute and 45 seconds, which equates to a total of 105 seconds. To find the current linear speed, we need to determine the wheel’s radius and then compute the arc length covered in one revolution, converting that to miles per hour.
From the problem, the Ferris wheel matches the size of Chicago’s original Ferris wheel, which had a diameter of approximately 250 feet. Assuming this standard size, the radius (r) is 125 feet.
The circumference (C) of the wheel, which is the path of a cart in one full revolution, is:
C = 2πr = 2 × 3.1416 × 125 ≈ 785.4 feet
The wheel completes one revolution in 105 seconds, so the linear speed v is:
v = distance / time = 785.4 feet / 105 sec ≈ 7.48 feet/sec
Converting feet/sec to miles/hour:
1 mile = 5280 feet
v = (7.48 ft/sec) × (3600 sec/hour) / 5280 ≈ 5.1 mph
Thus, the current speed of the cart is approximately 5.1 mph.
Calculating the Required Speed to Maximize Safety and Thrill
Regulations restrict maximum speed to 12 mph to ensure rider safety. To increase the speed while remaining within safety limits, the wheel's rotation time needs to be decreased.
Desired maximum speed:
v_max = 12 mph
In feet/sec:
v_max = 12 × 5280 / 3600 ≈ 17.6 ft/sec
The required circumference based on maximum speed:
C_new = v_max × time per revolution
From the circumference relation:
C_new = 785.4 feet (original size) is the current, but to reach 12 mph, the time per revolution must decrease:
time = circumference / max speed = 785.4 / 17.6 ≈ 44.6 seconds
This means the new rotation period should be approximately 44.6 seconds for the wheel to attain 12 mph at its current size.
Alternatively, if the size is to remain unchanged, the control panel should be adjusted to reduce the rotation period from 105 seconds to approximately 45 seconds, significantly increasing the ride’s thrill factor while adhering to safety standards.
Graphing the Height Function of the Carousel
The height h(t) of a cart over time can be modeled by a cosine function:
h(t) = r × cos(ωt + φ) + h₀
Where:
- r = radius of the wheel (125 ft)
- ω = angular speed = 2π / T, with T = rotation period in seconds
- φ = phase shift depending on the starting position
- h₀ = vertical offset, which in this context is the height of the center of the wheel
Given that the cart oscillates up and down over a period T, with four oscillations, the frequency f is 4 per cycle.
The graph will show the height starting from the bottom, reaching the maximum at 20 feet above the ground, corresponding to the top position of the ride. To model this accurately, we need to factor in the initial phase and the height at time t=0.
Assuming the initial height is at the bottom, with the center of the wheel at the midpoint of the height range:
h(t) = 125 × cos(ωt) + h₀
Where:
- h₀ = height of the wheel's center, which adds to the vertical offset
- ω = 2π / T
Given that T is 45 seconds (from the previous calculation), then:
ω = 2π / 45 ≈ 0.1396 rad/sec
The maximum height from this model occurs when cos(ωt) = 1, so maximum height h_max = 125 + h₀, and minimum when cos(ωt) = -1, h_min = -125 + h₀. Setting h_max to match the topmost position above ground (say, the maximum height the cart reaches), we could solve for h₀ based on the initial position.
The graph of this function will demonstrate the oscillation, with peaks aligned with the time the rider reaches 20 feet above ground. Calculating the specific t values when the height is 20 feet involves solving:
20 = 125 × cos(ωt) + h₀
Assuming h₀ is set to match the average height, and adjusting phase accordingly, the timing can be mapped for camera activation.
Timing the Camera for Photographs
The camera is set to take pictures when the cart reaches 20 feet on both ascent and descent. Using the height function h(t), and solving for t when h(t) = 20 feet:
20 = 125 × cos(ωt) + h₀
Suppose h₀ is approximately 125 feet (assuming center height). Rearranged:
cos(ωt) = (20 - h₀) / 125
Given the previous assumptions, if h₀ ≈ 125, then:
cos(ωt) = (20 - 125) / 125 = -105 / 125 ≈ -0.84
Now, solving for t:
t = (1 / ω) × arccos(-0.84) ≈ (1 / 0.1396) × 2.553 ≈ 18.29 seconds
The camera should be programmed to activate approximately 18 seconds after the ride starts, both on the oncoming and receding sides, to capture images at 20 feet height.
Timing the Water Spray
The water spray is to activate after 15 seconds into the ride, but the timing depends on the cart’s position at that moment. Using the height function, we check if the cart is near 20 feet at t=15 seconds:
Calculate h(15):
h(15) = 125 × cos(0.1396 × 15) + h₀
cos(0.1396 × 15) = cos(2.094) ≈ -0.5
Therefore:
h(15) = 125 × (-0.5) + h₀ = -62.5 + h₀
If h₀ ≈ 125, then:
h(15) ≈ 62.5 feet
This indicates the cart is well above the 20 feet height at 15 seconds, meaning the spray can safely activate without conflicting with the camera shot.
Moreover, the water feature timing can be synchronized with the oscillation phase so that the water is sprayed when the cart is at a suitable height and position, avoiding interference with the camera at 20 feet.
Summary and Engineering Recommendations
To improve the Ferris wheel's safety and ride quality, the control panel speed should be increased from approximately 5.1 mph to a maximum of 12 mph, which will require reducing the rotation period from 105 seconds to roughly 45 seconds. This adjustment heightens the thrill factor while still complying with safety regulations, as the maximum speed remains within the permissible 12 mph.
The height of each cart can be modeled by a cosine function with a period of 45 seconds, allowing precise timing for camera shots at 20 feet. The cameras should be triggered approximately 18 seconds into the ride, on both ascent and descent, to capture the riders at the desired height.
Timing the water spray is feasible 15 seconds into the ride, given the oscillation phase, and will not interfere with the camera shots if properly synchronized. The water feature should be activated when the cart is higher than 20 feet, ensuring safety, clear photos, and rider enjoyment.
Overall, these adjustments will enhance the ride's excitement level, ensure rider safety, and provide memorable visual captures for visitors.
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