My answer:
Y= -82.5 cos (4pi/3)x+91.5
(Period: 1.5=2pi/b --> b= 4pi/3 )
The diameter of the wheel is 165 feet, it rotates at 1.5 revolutions per minute, and the bottom of the wheel is 9 feet above the ground. Find an equation that gives a passenger's height above the ground at any time t during the ride. Assume the passenger starts the ride at the bottom of the wheel.
Looks good to me.
I had done this for you on Tuesday and had verified it for you at the end of the solution
http://www.jiskha.com/display.cgi?id=1374031966
To find an equation that gives a passenger's height above the ground at any time during the ride, we can use a sinusoidal function. Since the wheel completes one revolution when the passenger reaches the same position again, the function will have a period of one revolution.
Let's break down the given information to determine the constants in our equation:
1. The diameter of the wheel is 165 feet, which means the radius is half of that, or 82.5 feet. The cosine function typically oscillates between -1 and 1, so we'll multiply our equation by 82.5 to scale it correctly.
2. The wheel rotates at 1.5 revolutions per minute. Since the period is the time it takes for one revolution, we can calculate the period by dividing 1 by the number of revolutions, which gives us a period of 2/3 minutes or 1.5 minutes (1.5 = 2pi/b, solve for b).
3. The bottom of the wheel is 9 feet above the ground, which means this will be our vertical shift in the equation.
Putting it all together, we can use the following equation:
y = A cos(Bx) + C
where:
- A represents the amplitude of the function, which is the maximum displacement from the vertical shift. In this case, A = 82.5.
- B determines the frequency or how many cycles occur in a given interval. In this case, B = 2π/period = 2π/1.5.
- C is the vertical shift, which in this case is 9 feet.
Substituting these values into the equation, we get:
y = 82.5 cos((2π/1.5)x) + 9
Simplifying further, we have:
y = -82.5 cos(4π/3)x + 91.5
So, the equation that gives a passenger's height above the ground at any time (t) during the ride is y = -82.5 cos(4π/3)x + 91.5.