Brian is riding a Ferris wheel. The wheel has a radius of 25 feet, and at his lowest point, Brian is 8 feet off the ground. Brian times how long it takes to travel from the lowest point to the highest point and finds that it takes 8 seconds. Write a sinusoidal equation to model Brian’s movement around the Ferris wheel.

I will use a sine curve

amplitude = 25
so let's start with y = 25 sin t , where t is in seconds
period = 16 seconds
2π/k = 16
16k = 2π
k = π/8
so far y = 25 sin π/8 t
we want the lowest point to be +8, so we have to raise the sine curve 33 units
so far: y = 25 sin π/8 t + 33

we want our lowest point to be +8 when t = 0
so we need a phase shift
y = 25 sin π/8(t + k)+ 33
8 = 25 sin (π/8)k + 33
-25 = 25 sin πk/8
sin πk/8 = -1
I know sin 3π/2 = -1
so πk/8 = 3π/2
πk = 12π
k = 12

y = 25 sin π/8(t + 12)+ 33
min = 8, max = 58, half way = 33

check at the main 4 intervals of a period
t = 0 ---> y = 25 sin(3π/2) + 33 = 8 , good
t = 4 ---> y = 25 sin(2π) + 33 = 33 , half way, good
t = 8 ---> y = 25sin (5π/2) + 33 = 58, good
t = 12 --> y = 25sin(3π) + 33 = 33 , good, coming back down
t = 16 --> y =25sin(7π/2) + 33 = 8 , back to the bottom,
ALL IS GOOD!

Thanks....got it

To model Brian's movement around the Ferris wheel using a sinusoidal equation, we can start by analyzing the characteristics of the given situation. Let's consider the height of Brian at any given time.

We know that the Ferris wheel has a radius of 25 feet, so we can assume that the height varies between -25 and 25 feet. Furthermore, at his lowest point, Brian is 8 feet off the ground, so the vertical shift of our sinusoidal equation will be 8 units.

Next, we need to determine the period of the movement, which is the time it takes for one complete cycle. We are given that it takes Brian 8 seconds to travel from the lowest point to the highest point, which represents one cycle. Therefore, the period of our sinusoidal equation is 8 units.

We can use the standard form of the sinusoidal equation, which is:
y = A sin(B(x - C)) + D

Based on the information we have so far, we can conclude that:
- A represents the amplitude (half the vertical distance between the highest and lowest points): A = 25 - (-25) = 50
- B represents the frequency (number of cycles within a specific period): B = 2π/period = 2π/8 = π/4
- C represents the phase shift (horizontal shift): We can assume C = 0 since we have not been given any information about a shift.
- D represents the vertical shift: D = 8

Substituting the values into the equation, we have:
y = 50 sin(π/4(x - 0)) + 8

Simplifying this equation, we get:
y = 50 sin(πx/4) + 8

Therefore, the sinusoidal equation to model Brian's movement around the Ferris wheel is:
y = 50 sin(πx/4) + 8