The wheel on a child's bicycle is 48 cm in diameter. To test the strength of the tire, William lifts the back wheel 1 m from the ground and spins it at 36 turns/min. He attached a blue sticker to the bottom of the tire beforehand. Assuming that William spins the tire at a constant rate, find the rule for a sine function that calculates the height of the blue sticker (in cm) in terms of the elapsed time in seconds.
Please help :)
To find the rule for a sine function that calculates the height of the blue sticker in terms of the elapsed time, let's break down the problem step by step:
Step 1: Calculate the circumference of the wheel
The circumference of a circle is calculated using the formula: C = π * d, where C is the circumference and d is the diameter. In this case, the diameter is given as 48 cm, so we can calculate the circumference as follows:
C = π * 48 cm
C ≈ 150.796 cm
This means that when the wheel completes one revolution, the distance covered by the blue sticker is approximately 150.796 cm.
Step 2: Convert the given speed of rotation to cm/sec
The speed of rotation is given as 36 turns/minute. Since we want the answer in cm/sec, we need to convert minutes to seconds. There are 60 seconds in a minute, so the conversion is as follows:
36 turns/minute = 36 turns/60 seconds
= 0.6 turns/second
So, the wheel completes 0.6 revolutions in one second.
Step 3: Write the rule for the sine function
The height of the sticker can be represented using a sine function because it moves up and down in a periodic manner. The general formula for a sine function is:
y = A * sin(B * t + C) + D
In this case:
- A represents the amplitude, which is half the distance between the highest and lowest points.
- B represents the frequency, which is related to how fast the function repeats.
- C represents the phase shift, which determines the horizontal displacement of the function.
- D represents the vertical displacement of the function.
Step 4: Determine the values for A, B, C, and D
- A: The amplitude is half the distance covered by the sticker in one revolution, which is half the circumference of the wheel. So, A = 0.5 * 150.796 cm ≈ 75.398 cm.
- B: The frequency is the number of cycles completed in one second. Since the wheel completes 0.6 revolutions in one second, B = 2π * 0.6 ≈ 3.7699.
- C: The phase shift is 0 because we start measuring from the lowest point.
- D: The vertical displacement is the height from which the wheel is lifted, which is 100 cm.
Therefore, the rule for the sine function is:
y = 75.398 * sin(3.7699 * t) + 100
In this equation, t represents the elapsed time in seconds and y represents the height of the blue sticker in cm.