Suppose that a water wheel has a radius of 22 feet and makes a complete revolution every 14 seconds. The bottom of the water water wheel is 1 foot above the ground level. One step of the water wheel is painted red to be able to determine the amount of time for one complete revolution. State the function that represent the height of the painted step on the wheel after t seconds. Assume that the painted step is at the top of the wheel when t=0.
period=pi/7
wheel radius=22 means the amplitude is 22. So, y is something like
y = A+22sin(x)
Since the bottom is at y=1, that means
y = 23 + 22sin(x)
Oops. The wheel starts at the top when t=0, so we really have
y = 23 + 22cos(kx)
The period is 14, so
2pi/k = 14
k = pi/7
y = 23 + 22cos(pi/7 x)
The height of the painted step on the wheel after t seconds can be represented by the function:
h(t) = 22 + 22 sin((2π/14)t + π/2)
In this function, the term "22" represents the radius of the water wheel, and the term "22 sin((2π/14)t + π/2)" represents the vertical displacement of the painted step. The period of the function is given by 2π/B, where B is the coefficient of t in the function. In this case, B = 2π/14, so the period is π/7.
To find the function that represents the height of the painted step on the water wheel, we need to use the equation of a circle.
The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
In this case, the water wheel has a radius of 22 feet, and the bottom of the water wheel is 1 foot above the ground level. Let's assume that the center of the wheel is at the origin (0,0) of a coordinate system.
Since the wheel makes a complete revolution every 14 seconds, we can calculate the period of the revolution using the formula:
period = 2 * pi / (angular velocity)
Given that period = pi/7, we can solve for the angular velocity:
pi/7 = 2 * pi / (angular velocity)
Cross-multiplying and simplifying:
angular velocity = 2 * pi * 7 / pi = 14 radians per second
Now, to find the function that represents the height of the painted step on the wheel after t seconds, we need to determine the angle at which the step is located as time progresses.
Since the wheel is rotating at a constant angular velocity, we can use the equation:
angle = angular velocity * t
Substituting the value of the angular velocity, we have:
angle = 14t
Now, we can plug this angle into the equation of the circle. Since the center of the wheel is at (0,0), the equation simplifies to:
x^2 + y^2 = 22^2
Since the step is painted on the wheel, the x-coordinate will always be zero, so we can simplify further:
0^2 + y^2 = 22^2
Simplifying:
y^2 = 22^2
y = +/- 22
However, we know that the bottom of the wheel is 1 foot above the ground level, so we should take the positive value:
y = 22
Therefore, the function that represents the height of the painted step on the wheel after t seconds is:
f(t) = 22 feet