QuestionFind the height $h$ of a passenger on a Ferris wheel (radius $10 \mathrm{~m}$ , revolution time $48 \mathrm{~s}$ ) at time $t$ .

Studdy Solution

STEP 1

Assumptions1. The radius of the Ferris wheel is10 m. . The Ferris wheel completes one revolution every48 seconds.
3. The lowest point of the ride is m above the ground.
4. The ride starts from the bottom of the wheel.
5. We are looking for a function that represents the height above the ground at any time.

STEP 2

We need to understand that the height of the passenger at any time is a sinusoidal function because the Ferris wheel moves in a circular motion. The general form of a sinusoidal function is $h(t) = A \sin(B(t - C)) + D$ where- A is the amplitude (half the distance between the maximum and minimum values of the function, which is the radius of the Ferris wheel in this case) - B is the frequency (how often the Ferris wheel completes a full cycle, which is related to the time it takes to complete one revolution) - C is the horizontal shift (how much the graph is shifted horizontally, which is related to when the ride starts) - D is the vertical shift (how much the graph is shifted vertically, which is related to the height of the lowest point of the ride)

STEP 3

We can start by finding the values of A, B, C, and D for our function.
A is the radius of the Ferris wheel, which is10 m.
B is related to the time it takes to complete one revolution. Since the Ferris wheel completes one revolution every48 seconds, the frequency is $\frac{1}{48}$ revolutions per second. However, since the sine function completes one cycle every $2\pi$ units, we need to multiply the frequency by $2\pi$ to get B. So, B = $2\pi \times \frac{1}{48}$ .
C is related to when the ride starts. Since the ride starts from the bottom of the wheel, there is no horizontal shift, so C =0.
is the height of the lowest point of the ride, which is2 m.

STEP 4

Now, we can write the equation for the function that represents the height above the ground at any time $h(t) =10 \sin\left(2\pi \frac{1}{48} t\right) +2$ This is the equation that represents the height of the passenger above the ground at any time during the ride.

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