PRECALCULUS HELP IM BEGGING YOU Dean has built a ferris wheel in his backyard. It has a diameter of 10 feet and completes 30 revolutions per minute. Sketch a quick graph and find the period of the function that describes the height above the ground of a seat on the outside edge of this ferris wheel.

Period = 2 seconds

Graph is shown below

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Explanation:

1 minute = 60 seconds

We have 30 revolutions every 60 seconds, so we can write the ratio

30 rev: 60 sec

Divide both parts by 30 to turn that "30 rev" into "1 rev"

30 rev: 60 sec

30/30 rev: 60/30 sec

1 rev : 2 sec

This tells us that each full revolution takes 2 seconds. This is the period of the ferris wheel. The period is the length of each cycle.

The cyclic or periodic nature of this motion heavily implies we'll be dealing with a sine or cosine function of some kind. Let's go with sine.

Side note: cosine is really a sine function in disguise (it's just a phase shifted version of it).

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The general sine curve equation is

y = A*sin(B(x-C)) + D

where

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The wheel is 10 feet in diameter, so 10/2 = 5 is the radius. This is also the amplitude because it is the distance from the midline to either the highest point (10 ft) or lowest point (0 ft). So A = 5.

Since T = 2 is the period, this means,

B = 2pi/T

B = 2pi/2

B = pi

We don't have any phase shifts to worry about so C = 0.

The midline is D = 5 since this is halfway up

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Putting all the pieces together, we get this sine equation

y = A*sin(B(x-C)) + D

y = 5*sin(pi*(x-0)) + 5

y = 5*sin(pi*x) + 5

The graph is shown below.

x = time in seconds, y = height of the seat

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