A Ferris wheel has a radius of 40 feet. The bottom of the Ferris wheel sits 0.7 feet above the ground. You board the Ferris wheel at the 6 o'clock position and rotate counter-clockwise.
a.) Write a function f that determines your height above the ground (in feet) in terms of the number of radians you have swept out from the 6 o'clock position, a.
f(a)=_________
b.) Write a function g, that determines your height above the ground (in feet) in terms of the number of feet you have traveled since you started rotating, s.
g(s)=_________
Missing information:
how long is one rotation ?
a) To determine your height above the ground in terms of the number of radians you have swept out from the 6 o'clock position, we can use trigonometry.
Let's start by calculating the angle θ (in radians) that corresponds to a given arc length s. We know that the arc length s is related to the radius r and the angle θ by the equation:
s = r * θ
In this case, the radius of the Ferris wheel is 40 feet. The arc length s is the distance you have traveled since you started rotating, which is related to the angle a by the equation:
s = r * a
Substituting the values, we have:
s = 40 * a
Solving for a, we get:
a = s / 40
Now, let's determine the height h (in feet) as a function of the angle θ. We need to find the vertical component of the position on the Ferris wheel. Considering that the bottom of the Ferris wheel sits 0.7 feet above the ground, the height h can be calculated using the equation:
h = r * sin(θ) + 0.7
Substituting the value of θ in terms of a, we have:
h = 40 * sin(a) + 0.7
Therefore, the function f that determines your height above the ground (in feet) in terms of the number of radians you have swept out from the 6 o'clock position is:
f(a) = 40 * sin(a) + 0.7
b) To determine your height above the ground in terms of the number of feet you have traveled since you started rotating, we can use the relationship between arc length and angle again.
From the previous equation s = r * a, we can express angle a in terms of arc length s:
a = s / 40
Now, we need to find the height h as a function of the arc length s. We use the same equation as in part (a):
h = 40 * sin(a) + 0.7
Substituting the value of a, we get:
h = 40 * sin(s / 40) + 0.7
Therefore, the function g that determines your height above the ground (in feet) in terms of the number of feet you have traveled since you started rotating is:
g(s) = 40 * sin(s / 40) + 0.7