Rachel is bowling with her friends. Her bowling ball has a radius of 4.1 inches. As she bowls she tracks the location of the finger hole above the ground. She starts tracking the location when the finger hole is at the 12 o'clock position and she notices that she got some backspin on the ball and it rotates counter-clockwise.
Write a function f that determines the height of the finger hole above the ground (in inches) in terms of the number of radians a the ball has rotated since she started tracking the finger hole. (Note that a is a number of radians swept out from the 12-o'clock position.)
f(a)=
f(a) = "The height of the finger hole above the ground is directly proportional to the number of radians the ball has rotated, but inversely proportional to the radius of the ball. Let's calculate it using some clown math!"
f(a) = (a / 4.1) inches
Disclaimer: Clown math may not always be accurate, so please use with caution!
To determine the height of the finger hole above the ground (in inches) in terms of the number of radians a the ball has rotated, we can use the formula for the circumference of a circle:
Circumference = 2 * π * radius
Since the finger hole is on the circumference of the ball, its height above the ground can be calculated using trigonometry.
Let h be the height of the finger hole above the ground.
First, we need to find the angle θ that the finger hole has rotated. Since the ball started at the 12 o'clock position and rotates counter-clockwise, we can relate a to θ as follows:
θ = π/2 - a
Next, we can calculate the arc length s from the 12 o'clock position to the finger hole:
s = radius * θ
Then, we can use trigonometry to find the height of the finger hole:
h = radius * sin(θ)
Combining these equations, we can define the function f(a) as follows:
f(a) = radius * sin(π/2 - a)
To determine the height of the finger hole above the ground in terms of the number of radians the ball has rotated, we can use trigonometry.
Let's assume that the height of the finger hole at the 12 o'clock position is h0.
As the ball rotates counter-clockwise, the finger hole follows a circular path. The height of the finger hole can be represented by the vertical displacement from the position at 12 o'clock.
We know that the radius of the ball is 4.1 inches. Let's call the vertical displacement from the 12 o'clock position h.
When the finger hole has rotated by an angle of a radians, we can use the cosine function to find the height h:
h = r * cos(a)
where r is the radius of the ball and a is the angle rotated.
Therefore, the function f(a) that determines the height of the finger hole above the ground in terms of radians can be expressed as:
f(a) = 4.1 * cos(a)
This function will give you the height of the finger hole above the ground based on the number of radians the ball has rotated since tracking started.
if the period is k seconds, then
f(a) = r cos(2π/k a)
review the sine/cosine functions