1. A Ferris wheel with a radius of 7m makes one complete revolution every 16 s. The bottom of the wheel is 1.5m above ground.

a)Find the equation of the graph
b)predict how the graph and the equation will change if the Ferris wheel turns more slowly
c) test your predictions from part b by drawing a graph for three revolutions and finding a equation if the wheel completes one revolution every 20 s.

since sin(kt) has period 2pi/k, we have

y = 7sin(pi/8 t) + 8.5

with appropriate shift depending on where the wheel is at time 0.

slower turning means longer period, so the coefficient of t will decrease.

why pi?

set 2pi/k = 16 and solve for k.

Because the period is 16. Normally, sin(x) has period 2pi. Review your text and reread my first line.

i dont know the answer

a) Equation: Let's assume that the center of the Ferris wheel is at the origin of the coordinate system. The equation of the graph can be described using the cosine function.

The general equation for a graph of a Ferris wheel with radius r and the bottom of the wheel at a height h above the ground is:

y = r*cos(θ) + h

In this case, the radius is 7m and the bottom of the wheel is at a height of 1.5m, so the equation becomes:

y = 7*cos(θ) + 1.5

b) If the Ferris wheel turns more slowly, it will take a longer time to complete one revolution. This means that the period (T) of the cosine function will increase. As a result, the graph will stretch horizontally and the peaks and valleys will become less frequent.

c) To test our predictions, let's draw a graph for three revolutions (3*2π) and find the equation if the wheel completes one revolution every 20 seconds.

Assuming the same radius of 7m and bottom height of 1.5m, the new equation can be written as:

y = 7*cos((2π/20)t) + 1.5

By graphing this equation for three revolutions, we can observe the changes in the shape and frequency of the graph.

a) To find the equation of the graph for the Ferris wheel, we can model its motion using a sinusoidal function. The general form of a sinusoidal function is given by:

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

Where:
A represents the amplitude of the function (half the difference between the maximum and minimum values),
B determines the frequency or how often the function repeats over a given interval,
C represents the horizontal shift or phase shift of the function, and
D is the vertical shift or the Midline (average) value of the function.

For the Ferris wheel, we know:
- The radius is 7m, which is the amplitude of the sinusoidal function.
- The wheel completes one revolution every 16s, so the period of the function is 16s.

The period of a sinusoidal function is given by:
Period = 2π/B

From this, we can find the value of B using:
B = 2π/Period = 2π/16s = π/8

Thus, the equation of the graph for the Ferris wheel can be written as:
y = 7 * sin((π/8)(x - C)) + D

b) If the Ferris wheel turns more slowly, the period of the function will increase. Let's say the new period is T seconds. Following the same logic as above, the value of B will change as:
B = 2π/T

Because the radius (amplitude) remains the same, the equation of the graph will become:
y = 7 * sin((2π/T)(x - C)) + D

c) Let's test our predictions from part b by drawing a graph for three revolutions, where the wheel completes one revolution every 20s.

In this case, the period T = 20s, and we can find the value of B as:
B = 2π/20s = π/10

Using this information, the equation of the graph for the Ferris wheel for three revolutions (x values from 0 to 60) becomes:
y = 7 * sin((π/10)(x - C)) + D

The exact values of C and D will depend on the starting position and the midline of the Ferris wheel.