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 height above water of the initial point after t seconds.
a. h=8cos((2π)/2 t)+7
b. h=-8cos3t+7
c. h=-8cos((2π)/3 t)+7
d. h=8cos3t+7
I think this is C?
2. use your function to find the height of the initial point after 5.5 seconds.
a. 7.5 ft
b. 11 ft
c. 10.4 ft
d. 6.5 ft
The period is clearly 60/20 seconds, or 3 seconds.
the cosine function should be of the form
cos(2PI/period* t)
C. is correct.
so would 2. be C as well?
Yes, you are correct for question 1. The correct answer is (c) h = -8cos((2π)/3 t) + 7.
To solve question 2 using the function h = -8cos((2π)/3 t) + 7, substitute t = 5.5 seconds into the equation:
h = -8cos((2π)/3 * 5.5) + 7
Calculating this expression, we get:
h ≈ -8cos(11π/3) + 7
Since cos(11π/3) is approximately -0.5, we can substitute this value into the equation:
h ≈ -8(-0.5) + 7
Simplifying, we get:
h ≈ 4 + 7
h ≈ 11 ft
Therefore, the correct answer for question 2 is (b) 11 ft.
To answer these questions, we need to understand the relationship between time (t) and the height (h) of the initial point on the paddle wheel.
1. The height of the initial point can be modeled with a cosine function because the paddle wheel rotates in a circular motion. The general form of a cosine function is:
h = A * cos(B * (t - C)) + D
Where A represents the amplitude (the maximum height), B represents the angular frequency (how quickly the function oscillates), C represents the phase shift (the initial position), and D represents the vertical shift (the average height).
In this case, the amplitude is |A| = 8 (since the paddle wheel has a maximum depth of 1 foot and the highest point would be 8 feet above the water), and the phase shift is 0 (since the point is located at the lowest point of the wheel at t=0). Therefore, the correct function is:
h = 8 * cos((2π)/2 * t) + 7
So, the answer is (a) h = 8cos((2π)/2 t) + 7.
2. To find the height of the initial point after 5.5 seconds, we can substitute t = 5.5 into the function h = 8 * cos((2π)/2 * t) + 7:
h = 8 * cos((2π)/2 * 5.5) + 7
Evaluating this expression, we find:
h ≈ 7.5 ft
Therefore, the answer is (a) 7.5 ft.