a ferris wheel is 50 meters in diameter and boarded from a platform that is 4 meters above the ground. the…
a ferris wheel is 50 meters in diameter and boarded from a platform that is 4 meters above the ground. the six oclock position on the ferris wheel is level with the loading platform. the wheel completes 1 full revolution in 4 minutes. the function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. what is the amplitude? meters what is the midline? y = meters what is the period? minutes how high are you off of the ground after 2 minutes? meters question help: video
Answer
Amplitude
Explanation:
Step1: Recall Amplitude Definition
Amplitude of a sinusoidal function (like the height on a Ferris wheel) is half the distance between the maximum and minimum values, or equivalently, the radius of the Ferris wheel.
Step2: Calculate Amplitude
The diameter of the Ferris wheel is 50 meters, so the radius (which is the amplitude) is $\frac{50}{2} = 25$ meters.
Answer:
25
Midline
Explanation:
Step1: Recall Midline Definition
The midline of a sinusoidal function representing height on a Ferris wheel is the average of the maximum and minimum heights, or the height of the center of the Ferris wheel above the ground.
Step2: Calculate Midline
The center of the Ferris wheel is at a height equal to the height of the platform plus the radius of the Ferris wheel. The platform is 4 meters above the ground, and the radius is 25 meters, so the midline $y$ is $4 + 25 = 29$ meters.
Answer:
29
Period
Explanation:
Step1: Recall Period Definition
The period of a periodic function (like the Ferris wheel's rotation) is the time it takes to complete one full revolution.
Step2: Determine Period
The problem states the wheel completes 1 full revolution in 4 minutes, so the period is 4 minutes.
Answer:
4
Height after 2 minutes
Explanation:
Step1: Analyze Rotation at t = 2
Since the period is 4 minutes, at $t = 2$ minutes, the Ferris wheel has completed half a revolution (because $2=\frac{4}{2}$).
Step2: Determine Height
Starting from the 6 - o'clock position (level with the platform, 4 meters above ground), after half a revolution, we will be at the 12 - o'clock position. The height at the 12 - o'clock position is the midline height plus the amplitude (or the height of the center plus the radius). The midline is 29 meters and the amplitude is 25 meters, so the height is $29+25 = 54$ meters. Alternatively, using the function: the general form of the height function for a Ferris wheel is $h(t)=A\cos(\omega t)+k$ (or sine, but cosine is convenient here as we start at the minimum). The amplitude $A = 25$, the midline $k = 29$, and the angular frequency $\omega=\frac{2\pi}{T}=\frac{2\pi}{4}=\frac{\pi}{2}$. So $h(t)=25\cos\left(\frac{\pi}{2}t\right)+29$. At $t = 2$, $h(2)=25\cos\left(\frac{\pi}{2}\times2\right)+29=25\cos(\pi)+29=25\times(- 1)+29 = 4$? Wait, no, wait. Wait, when $t = 0$, we are at the 6 - o'clock position (minimum height, 4 meters). The cosine function starts at maximum, so maybe we should use a sine function or adjust the cosine. Wait, actually, when $t = 0$, $h(0)=4$. The center is at 29, amplitude 25, so the function can be written as $h(t)=- 25\cos\left(\frac{\pi}{2}t\right)+29$ (because at $t = 0$, $\cos(0)=1$, so $-25\times1 + 29 = 4$, which is correct). Then at $t = 2$, $h(2)=-25\cos\left(\frac{\pi}{2}\times2\right)+29=-25\cos(\pi)+29=-25\times(-1)+29 = 25 + 29 = 54$ meters.
Answer:
54