A ferris wheel is 63 feet tall when it reaches its highest point at 8 seconds. At the bottom of the ride, you are 3 feet off the ground. So how high off the ground will you be 5 seconds after the ride starts?
Write a function f so that f(t) will equal how high in feet you will be off the ground t seconds later.
So I think the diameter is 60 feet meaning my radius is 30.And at the highest point, it takes me 8 seconds, so a full revolution must be 16 seconds. How do i make a equation to find how high I'll be 5 seconds later when I start from the bottom?
amplitude=30
Period=2π/16
Is this right?
since cos(kt) has its max at t=0, and you want your max at t=8, then using your correct amplitude and period, we have
y = 33 + 30cos((pi/8)(t-8))
So, now just plug in t=5 to answer the question.
You can see the graph at
http://www.wolframalpha.com/input/?i=33%2B30cos%28%28pi%2F8%29%28x-8%29%29
when i plug in t=5, I get 62.99 feet, but isn't that the height when t=8 seconds?
well, geez. when t=8, y=33+30cos(0) = 63
Isn't that what you wanted, the max height at t=8?
Sorry, I'll be more clear. The question already stated how high the ferris wheel is at t=8 seconds. The highest point the ferris wheel reaches is 63 feet and the time for that is 8 seconds. What I'm trying to find out is how high the ferris wheel is when t=5 seconds using an equation I can plug in.
John, I don't know how you got 63 when t=5
I got 44.48 when t = 5
We could have used a sine curve
height = 30sin ((πt/8)(t-4) ) +33
check:
when t = 0 , height = 3 , ok
when t = 8 , height = 63, ok
when t = 4, height = 4.48 , the same result we get when we use Steve's cosine function.
Yeah sorry, my bad. I must have plugged in the numbers wrong, since I just got 44.5 for my answer. You guys are right, Thanks so much Steve and Reiny for your help!
To find the equation for how high you will be off the ground at any given time, we can start by analyzing the motion of the ferris wheel.
The ferris wheel completes one revolution in 16 seconds, so the period would indeed be T = 16.
The amplitude of the ferris wheel's motion represents its maximum height, and in this case, the ferris wheel is 63 feet tall at its highest point. However, it is important to note that the amplitude should be measured from the center of the motion (in this case, the ground level) to the highest point. Therefore, the amplitude should be 63/2 = 31.5 feet.
Now that we have determined the amplitude (A) and period (T), we can use the standard equation for a sinusoidal function:
f(t) = A sin(2πt / T) + C
where f(t) represents the height off the ground at time t, and C represents any vertical shift (in this case, the initial height).
Since at the bottom of the ride, you are 3 feet off the ground, C would be 3.
Using these values, the equation for how high you will be off the ground at any given time is:
f(t) = 31.5 sin(2πt / 16) + 3
To find how high you will be 5 seconds after the ride starts, you can substitute t = 5 into the equation:
f(5) = 31.5 sin(2π(5) / 16) + 3
Now you can evaluate this expression to find the exact height off the ground at that specific time.