If a ferris wheel with a radius of 25 feet and a center at 30 feet above the ground is rotating at 0.25 radians per second, define a formula for a function g that represents Shawna's height above the ground (in feet) in terms of the number of seconds since the ferris wheel started to rotate if she starts at the 9 o'clock position.
The axle is at height 25+5=30, so since ω = 0.25,
g = 30+25sin(0.25 t)
To define a formula for the function g that represents Shawna's height above the ground in terms of the number of seconds since the ferris wheel started to rotate, we can use the equation of a circle.
The equation of a circle centered at (h, k) with radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
In this case, the circle is the ferris wheel, center (h, k) = (0, 30), and radius r = 25.
We can use polar coordinates to represent the position on the ferris wheel. The angle θ (in radians) represents the position of a point on the ferris wheel. At the 9 o'clock position, the angle θ = π/2.
The polar coordinates for a point on the circle are given by:
x = r * cos(θ)
y = r * sin(θ)
Substituting the values, we can rewrite the equation of the circle as:
(cos(θ))^2 * 25^2 + (sin(θ))^2 * 25^2 = (x - 0)^2 + (y - 30)^2
Simplifying the equation:
625 * (cos(θ))^2 + 625 * (sin(θ))^2 = x^2 + y^2 - 60y + 900
Since Shawna's height above the ground is represented by y, our equation becomes:
625 * (cos(θ))^2 + 625 * (sin(θ))^2 = g^2 - 60g + 900
Using the trigonometric identity cos^2(θ) + sin^2(θ) = 1, we can simplify further:
625 + 0 = g^2 - 60g + 900
g^2 - 60g + 275 = 0
This equation can be factored as:
(g - 25)(g - 35) = 0
Therefore, the formula for the function g that represents Shawna's height above the ground in terms of the number of seconds since the ferris wheel started to rotate is:
g = 25 or g = 35
It means that Shawna's height above the ground will be either 25 feet or 35 feet, depending on the time since the ferris wheel started to rotate.
To define a formula for a function g that represents Shawna's height above the ground (in feet) in terms of the number of seconds since the Ferris wheel started to rotate, we can break down the problem into two parts:
1. Find Shawna's position relative to the center of the Ferris wheel at any given time.
2. Convert the relative position to Shawna's height above the ground.
Let's start with the first part.
1. Finding Shawna's position:
Since Shawna starts at the 9 o'clock position, her starting position can be represented as an angle of π/2 (90 degrees) counterclockwise from the positive x-axis. Let t represent the number of seconds since the Ferris wheel started to rotate.
The angular position of Shawna at any time t can be calculated using the equation:
θ(t) = π/2 - 0.25t
Here, θ(t) represents the angle in radians.
2. Converting the relative position to height above the ground:
To determine Shawna's height above the ground based on her angular position, we can use the equation of a circle.
Let h(t) represent Shawna's height above the ground at time t.
The equation of a circle is given by:
x² + y² = r²
where x and y represent the coordinates of a point on the circle's circumference and r is the radius.
In this case, Shawna's coordinate x will be the distance from the center of the Ferris wheel to Shawna, which is 30 feet. The coordinate y will represent her height above the ground.
We can express the height above the ground in terms of the radius and the angle by:
y = r * sin(θ(t))
Substituting the values, we have:
y = 25 * sin(π/2 - 0.25t)
Therefore, the formula for a function g that represents Shawna's height above the ground (in feet) in terms of the number of seconds since the Ferris wheel started to rotate is:
g(t) = 25 * sin(π/2 - 0.25t)