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
w = angular rate = 20*2pi/60 = 2pi/3 radians/sec
r = 16/2 = 8 feet
height of point above axle of wheel = -8cos wt
height of point above water = h = 7 - 8 cos wt
since w = 2 pi/3
h = 7 - 8 cos (2 pi t /3)
if t = 5.5
h = 7 - 8 cos (11 pi/3)
= -.839 ft
I do not understand what 3) has to do with this.
sin anything is -1 for 3pi/2, 3pi/2 + n*2pi
1) To write a cosine function with a phase shift of 0, we can start with the general form of a cosine function:
f(t) = A * cos(B * (t - C)) + D
Where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.
In this case, the amplitude is half the maximum depth of the paddle wheel, which is 1/2 foot. The frequency can be determined using the formula:
frequency = (2π) / period
Since the paddle wheel is revolving at a rate of 20 rpm, the period is 1 / 20 minutes. We convert this to seconds by multiplying by 60:
frequency = (2π) / (1 / 20 * 60) = 2π * 20 * 60 = 240π
The phase shift is given as 0, so C = 0.
Finally, the vertical shift is the height at the initial point, which is 0.
Therefore, the cosine function for the height above water at the initial point after t seconds is:
f(t) = (1/2) * cos(240πt)
2) To find the height of the initial point after 5.5 seconds, we can substitute t = 5.5 into the cosine function:
f(5.5) = (1/2) * cos(240π * 5.5)
Using a calculator to evaluate this expression:
f(5.5) ≈ 0.5
So, the height of the initial point after 5.5 seconds is approximately 0.5 feet above the water.
3) Now let's find the values of x for which the equation sin(x) = -1 is true.
Since the sine function takes values between -1 and 1, we look for the angles where sin(x) is equal to -1. This occurs at angles where the sine function reaches its minimum value.
The minimum value of sine function occurs at x = -π/2. However, we are looking for the solutions where sin(x) = -1, so we need to add an odd multiple of π to -π/2.
Therefore, the solutions to sin(x) = -1 are:
x = -π/2 + π * n, where n is an integer.