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
h=-8cos(2pie/3t)+7
11 feet
not sure
1) To write a cosine function with a phase shift of 0 for the height above water at the initial point after t seconds, we can use the equation:
h(t) = A * cos(B(t - C)) + D
Where:
A is the amplitude of the function (maximum height above water),
B is the period of the function (time taken to complete one full revolution of the paddle wheel),
C is the phase shift (shift in time),
D is the vertical shift (initial height above water).
In this case, the amplitude (A) is half the maximum depth of the paddle wheel under water, so A = 1/2 feet. The period (B) is related to the rate of revolution (RPM) by the formula B = (2π)/R, where R is the rate of revolution in revolutions per minute. So B = (2π)/20.
Since the phase shift is given as 0 (C = 0), and the initial height above water is 0 (D = 0), the cosine function becomes:
h(t) = (1/2) * cos((2π/20)t)
2) To find the height of the initial point after 5.5 seconds, we simply substitute t = 5.5 into the function:
h(5.5) = (1/2) * cos((2π/20) * 5.5)
Calculate this expression using a calculator or math software to get the numerical value.
3) To find the values of x for which the equation sin(x) = -1 is true, we need to determine the angles that have a sine value of -1. Recall that the sine function has a range of -1 to 1.
The angle at which sin(x) = -1 is -π/2 or -90 degrees. However, we need to find all the solutions, which have the same sine value but differ by multiples of 2π.
The general solution can be written as:
x = -π/2 + 2πn, where n is an integer
This captures all the angles that satisfy sin(x) = -1. To find specific values of x, substitute different values for n (e.g., -1, 0, 1, 2, ...) into the equation x = -π/2 + 2πn. Each time, calculate the corresponding value of x to get the solutions.