A Ferris wheel rotates around in 30 seconds. The maximum height above the ground is 55 feet, and the minimum height above the ground is 5 feet. Which function models the height as a function of
The Correct Answer and Explanation is :
To model the height of a Ferris wheel as a function of time, we can use a sinusoidal function. The general form of a sinusoidal function to model vertical motion is: h(t)=Asin(B(t−C))+Dh(t) = A \sin(B(t – C)) + D
Where:
- h(t)h(t) represents the height at time tt,
- AA is the amplitude (half the difference between the maximum and minimum height),
- BB controls the period of the function (how quickly it repeats),
- CC represents a horizontal shift (which we can set to zero for simplicity if the height starts at the minimum or maximum),
- DD represents the vertical shift (the middle of the range of heights).
Step 1: Finding the Amplitude
The amplitude AA is half the difference between the maximum and minimum heights of the Ferris wheel.
- Maximum height: 55 feet
- Minimum height: 5 feet
- The difference is 55−5=5055 – 5 = 50 feet.
- The amplitude is half of this difference: A=502=25 feetA = \frac{50}{2} = 25 \text{ feet}
Step 2: Finding the Vertical Shift
The vertical shift DD is the average of the maximum and minimum heights. D=55+52=30 feetD = \frac{55 + 5}{2} = 30 \text{ feet}
This means the center of the Ferris wheel is 30 feet above the ground.
Step 3: Finding the Period
The period of the function is the time it takes to complete one full rotation of the Ferris wheel. Since the Ferris wheel takes 30 seconds to complete one rotation, the period is 30 seconds.
The period of a sinusoidal function is given by 2πB\frac{2\pi}{B}, so: 2πB=30⇒B=2π30=π15\frac{2\pi}{B} = 30 \quad \Rightarrow \quad B = \frac{2\pi}{30} = \frac{\pi}{15}
Step 4: Writing the Function
Now that we have all the necessary components, the function becomes: h(t)=25sin(π15t)+30h(t) = 25 \sin\left(\frac{\pi}{15} t\right) + 30
This is the model for the height of the Ferris wheel at any time tt, where tt is in seconds.
Explanation:
- Amplitude: The amplitude of 25 feet reflects the distance from the middle of the Ferris wheel (30 feet) to the top (55 feet) and bottom (5 feet).
- Vertical Shift: The vertical shift of 30 feet centers the function so that the Ferris wheel oscillates symmetrically above and below this point.
- Period: The period of 30 seconds ensures that the function completes one full cycle (i.e., one full rotation) in the correct amount of time. The value B=π15B = \frac{\pi}{15} ensures this time period.
Thus, this function correctly models the height of a person on the Ferris wheel as a function of time.