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 represent the cosine function for the height above water, we need to consider the diameter, rate of revolution, and maximum depth of the paddle wheel. Let's break down the information given:
- Diameter of the paddle wheel = 16 feet, meaning the radius is half of that, so r = 8 feet.
- Rate of revolution = 20 rpm, which means the paddle wheel completes 20 full rotations in one minute.
- Maximum depth under water = 1 foot.
Knowing that the height above water can be represented as a cosine function, we can determine the formula by considering the properties of the cosine function:
y = A * cos(B(x - C)) + D
Where:
- A represents the amplitude (maximum value of the function) = radius of the paddle wheel = 8 feet.
- B represents the frequency (number of cycles in a given interval) = 1 rotation / 2π radians = 1/2π.
- C represents the phase shift (horizontal translation) = 0 seconds (initial point).
- D represents the vertical shift = maximum depth under water = -1 foot.
Plugging in the values, we get:
y = 8 * cos((1/2π)(x - 0)) + (-1)
y = 8 * cos((1/2π)x) - 1
Therefore, the cosine function with phase shift 0 for the height above water at the initial point after t seconds is given by y = 8 * cos((1/2π)t) - 1.
2) To find the height of the initial point after 5.5 seconds, we can substitute t = 5.5 into the function:
y = 8 * cos((1/2π) * 5.5) - 1
y ≈ 8 * cos(5.5/π) - 1
Evaluating this expression will give you the height of the initial point after 5.5 seconds.
3) To find the values of x for which the equation sin(x) = -1 is true, we need to consider the definition of sine function and its values:
- The sine function has a range of [-1, 1]. The value -1 indicates the minimum point.
- The sine function takes a value of -1 at two specific angles in a unit circle: -π/2 and 3π/2.
Therefore, we need to find the values of x where -π/2 and 3π/2 satisfy the equation sin(x) = -1.
Thus, the values of x for which the equation sin(x) = -1 is true are x = -π/2 and x = 3π/2.