The paddle wheel of a boat measures 16 feet in diameter and is revolving at a rate of 20 rpm. The maximum depth of the paddle wheel under water is 1 foot. Suppose a point is located at the lowest point of the wheel at t=0.
1) Write a cosine function with phase shift 0 for the height above water at the initial point after t seconds
2) use your function to find the height of the initial point after 5.5 seconds
3) and find the values of x for which the equation sin x= -1 is true.
Thanks for any assistance
1) To write a cosine function with a phase shift of 0 for the height above the water at the initial point after t seconds, we need to consider the relationship between the rotation of the paddle wheel and the height of the point.
The height of the point above the water can be modeled by a cosine function, where the amplitude is the maximum depth of the paddle wheel under the water (1 foot), the period is the time it takes for the wheel to make one complete revolution (60 seconds for 20 rpm), and the phase shift is 0 since we are starting at t=0.
The equation for the height above the water, h(t), can be represented as:
h(t) = A * cos(Bt)
where A is the amplitude (maximum depth), and B is the frequency (2π divided by the period).
In this case, A = 1 foot and the period is 60 seconds, so we have:
h(t) = 1 * cos((2π/60)t)
2) To find the height of the initial point after 5.5 seconds, we can substitute t = 5.5 into the cosine function we derived:
h(5.5) = 1 * cos((2π/60) * 5.5)
= 1 * cos(0.183π)
≈ 0.966 feet
So, the height of the initial point after 5.5 seconds is approximately 0.966 feet above the water.
3) To find the values of x for which the equation sin(x) = -1 is true, we can use the unit circle.
The sine function represents the y-coordinate on the unit circle, which ranges from -1 to 1. The value -1 corresponds to the points on the unit circle where the angle is π or 3π radians.
So, we have:
x = π + 2nπ or x = 3π + 2nπ
where n is an integer.
Therefore, the values of x for which sin(x) = -1 are x = π + 2nπ and x = 3π + 2nπ, where n is an integer.