Two Ferris wheels are side-by-side are rotating at the Math Fair. The first Ferris wheel has a radius of 7m and makes one complete revolution every 16 s. The bottom of the wheel is 1.5 m above ground. The second Ferris has a radius of 8m and completes one revolution every 20 s. The bottom of this wheel is 2 m above the ground. What are the equations of both Ferris wheels using sine as the base function? Graph each equation and estimate the times where the heights of a gondola are the same.

Question

19-07-2022
Mathematics

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Two Ferris wheels are side-by-side are rotating at the Math Fair. The first Ferris wheel has a radius of 7m and makes one complete revolution every 16 s. The bottom of the wheel is 1.5 m above ground. The second Ferris has a radius of 8m and completes one revolution every 20 s. The bottom of this wheel is 2 m above the ground. What are the equations of both Ferris wheels using sine as the base function? Graph each equation and estimate the times where the heights of a gondola are the same.

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MrRoyal MrRoyal · Answer 1 · 2022-07-21T14:52:54+02:00

The heights of the gondola are the same at 8.019 seconds and 26.446 seconds

How to determine the equation of the functions?

Ferris wheel 1

The given parameters are:

Radius, r = 7 m
Time, t = 16 s
Height above the ground, h = 1.5 m

The above means that:

Amplitude, A = 7
Period, T = 16
Minimum = 1.5

The sine function is represented as:

y = Asin(2π/T)t + c

Where

c = Amplitude - Minimum

c = 7 - 1.5

c = 5.5

So, we have:

y = 7sin(2πt/16) + 5.5

y = 7sin(πt/8) + 5.5

Shift to the left by π/2

y = 7sin(πt/8 - π/2) + 5.5

Ferris wheel 2

The given parameters are:

Radius, r = 8 m
Time, t = 20 s
Height above the ground, h = 2 m

The above means that:

Amplitude, A = 8
Period, T = 20
Minimum = 2

The sine function is represented as:

y = Asin(2π/T)t + c

Where

c = Amplitude - Minimum

c = 8 - 2

c = 6

So, we have:

y = 8sin(2πt/20) + 6

y = 8sin(πt/10) + 6

Shift to the left by π/2

y = 8sin(πt/10 - π/2) + 6

Hence, the equations of the wheels are y = 7sin(πt/8 - π/2) + 5 and y = 8sin(πt/10 - π/2) + 6

See attachment for their graphs

When they have the same heights

From the attached graph, we have:

f(x) = g(x) = 12.5 at x = 8.019

f(x) = g(x) = 9.511 at x = 26.446

Hence, the heights of the gondola are the same at 8.019 seconds and 26.446 seconds