A ferris wheel is 35 meters in diameter and boarded at ground level. The wheel makes one full rotation every 8 minutes, and at time (t=0) you are at the 3 o'clock position and descending. Let f(t) denote your height (in meters) above ground at t minutes. Find a formula for f(t).
I answered 17.5*sin((pi/4)*t)+17.5 but it's wrong I'm not sure what I'm doing incorrectly
Oh, a ferris wheel question! This is 'wheel'ly exciting! Let's get 'rolling' with the solution:
Since the ferris wheel makes one full rotation every 8 minutes, this means it completes 360 degrees in 8 minutes. So, every minute it moves 360/8 = 45 degrees.
At t=0, when you are at the 3 o'clock position and descending, you are 90 degrees below the horizontal line through the center of the ferris wheel. Since the diameter is 35 meters, that means you are 35 * sin(90) = 35 meters above the ground.
Now, every minute you move 45 degrees, so after t minutes, you would have moved 45t degrees. However, since you start from the 3 o'clock position, you are already at 90 degrees, so you need to subtract that from the total.
Putting it all together, the formula for your height above ground f(t) is:
f(t) = 35 * sin(45t - 90)
Keep in mind that f(t) gives the height above ground, so negative values mean you are below the ground level. Don't fall off, and always enjoy the 'heights' of the ferris wheel experience!
To find a formula for f(t), let's analyze the situation and break it down step by step.
1. First, let's determine the equation of the ferris wheel's circumference.
- The diameter of the ferris wheel is given as 35 meters, so the radius (r) is half of the diameter, which is 35/2 = 17.5 meters.
- The circumference of a circle is given by the formula C = 2πr, where π is a mathematical constant approximately equal to 3.14.
- Therefore, the circumference of the ferris wheel is C = 2π(17.5) = 35π meters.
2. Next, let's determine the period of one rotation of the ferris wheel.
- It is given that the ferris wheel makes one full rotation every 8 minutes.
3. Now, let's determine the average speed of the ferris wheel.
- If the circumference of the ferris wheel is covered in 8 minutes, the average speed is given by the formula V = S/T, where S is the distance traveled and T is the time taken.
- The distance traveled is the circumference of the ferris wheel, which we determined to be 35π meters.
- The time taken to cover this distance is 8 minutes.
- Therefore, the average speed is V = 35π / 8 meters per minute.
4. Next, let's determine the amplitude of the periodic motion.
- The ferris wheel starts at the ground level, so the initial height is zero.
- When the ferris wheel is at its highest point, you will reach a height equal to the radius, which is 17.5 meters.
- Therefore, the amplitude of the periodic motion is A = 17.5 meters.
5. Finally, let's put all the information together to find the formula for f(t) (your height above the ground at t minutes).
- Since the ferris wheel starts descending at t = 0 (3 o'clock position), we need to consider the downward motion.
- The equation that represents the height of the ferris wheel over time can be written as:
f(t) = A * sin(2π * t / T) + A
where:
- f(t) is your height above the ground at time t.
- A is the amplitude of the motion, which is 17.5 meters.
- t is the time in minutes.
- T is the period of one rotation, which is 8 minutes.
- sin represents the sine function, which gives the vertical position on the ferris wheel at a given angle.
- 2π * t / T determines the angle at which you are at a given time t.
Therefore, the formula for f(t) is:
f(t) = 17.5 * sin(2π * t / 8) + 17.5
You will need a phase shift
your amplitude is correct at 17.5
your vertical shift of 17.5 is also correct
period = 2π/k = 8
8k = 2π
k = 2π/8 = π/4
so your k for the period is correct
so let's adjust:
height = 17.5 sin (π/4)(t + d) + 17.5 , where d is a phase shift
So when you are at the 3:00 position and going downwards you must be 3/4 through a rotation and
t = 6 at a height of 17.5
sub in our equation:
17.5sin(π/4)(6+d) + 17.5 = 17.5
sin (π/4)(6+d) = 0
but I know that sin0 = 0 and sin π = 0
so (π/4)(6+d) = 0 or π/4(6+d) = π
d = -6
or
(1/4)(6+d) = 1
6+d = 4
d = -2
Just realized that if at 3:00 position you are going down, the wheel must be going clockwise (in math counterclockwise is a positive rotation)
But all is not lost, lets' just change our equation to
height = 17.5 sin (π/4)(-t + d) + 17.5
so our values of d would change:
d = 6 or d = 10
let's try d = 6
height = 17.5sin(π/4)(-t+6) + 17.5
testing:
t = 0 , height = 17.5 sin(π/4)(6) + 17.5 = 0 that's good
t - 2 , height = 17.5sin(π/4)(-2+6) + 17.5 = 17.5 , ok
t = 4 , height = 17.5sin(π/4)(-4+6) +17.5 = 35 , we are at the top
t = 6 , height = 17.5sin(π/4)(-6+6) +17.5 = 17.5 YEAHH, we are at 3:00 and coming back down
My equation is
f(t) = 17.5 sin (π/4)(-t+6) + 17.5
remember this equation is not unique,
if you recall sin(-x) = -sinx
so we could also write our equation as
f(t) = -17.5 sin (π/4)(t - 6) + 17.5