A Ferris wheel has a radius of 20 meters, makes a full rotation in 1 minute, and its axle stands 25 meters above the ground. The red chair starts at the top. We want to find: a) the period of the Ferris wheel. b) a sketch of two cycles of the graph of the red chair's height as a function of time, with labelled axes. c) the equation of the sine function describing the graph. d) the height of the chair after 20 seconds.

1. Problem Description

A Ferris wheel has a radius of 20 meters, makes a full rotation in 1 minute, and its axle stands 25 meters above the ground. The red chair starts at the top. We want to find:

a) the period of the Ferris wheel.

b) a sketch of two cycles of the graph of the red chair's height as a function of time, with labelled axes.

c) the equation of the sine function describing the graph.

d) the height of the chair after 20 seconds.

2. Solution Steps

a) The period of the Ferris wheel is the time it takes to complete one full rotation. The problem states that the Ferris wheel makes a full rotation in 1 minute. Since the units are in seconds, the period is 60 seconds.

b) The height of the axle is 25 meters. The radius of the Ferris wheel is 20 meters. The red chair starts at the top, so its initial height is $25 + 20 = 45$ meters. The minimum height is $25 - 20 = 5$ meters. The period is 60 seconds.

The graph should have the x-axis labelled "Time (seconds)" and the y-axis labelled "Height (meters)". The graph will start at (0, 45), reach (30, 5), and then (60, 45). Two cycles would extend to 120 seconds.

c) The general form of a sinusoidal function is:

$y = A \cos(B(x - C)) + D$

or

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

Since the red chair starts at the top, it's easier to use a cosine function. The amplitude $A$ is the radius, which is

2

0. The vertical shift $D$ is the height of the axle, which is

2

5. The period is 60, so $B = \frac{2\pi}{60} = \frac{\pi}{30}$.

Since we are using cosine and the chair starts at the top, the horizontal shift $C$ is

0. Therefore, the equation is:

$y = 20 \cos(\frac{\pi}{30} t) + 25$

d) To find the height of the chair after 20 seconds, plug $t=20$ into the equation:

$y = 20 \cos(\frac{\pi}{30} \cdot 20) + 25$

$y = 20 \cos(\frac{2\pi}{3}) + 25$

$y = 20 (-\frac{1}{2}) + 25$

$y = -10 + 25$

$y = 15$

3. Final Answer

a) 60 seconds

b) (Sketch: x-axis labelled "Time (seconds)", y-axis labelled "Height (meters)". Graph starts at (0, 45), reaches (30, 5), (60, 45), (90, 5), (120, 45))

c) $y = 20 \cos(\frac{\pi}{30} t) + 25$

d) 15 meters