A Ferris wheel at an amusement park reaches a maximum height of 50
metres and a minimum height of 4 metres. It takes 30 minutes for the wheel
to make one full rotation.
a. If a child gets on the Ferris wheel when ,
how high will he be after riding for 15 minutes?
b. Write a sine function to model the height of
the child minutes after boarding the Ferris wheel.
c. How high is the child if he has been riding
for 5 minutes
d. For what length to time will the child be
higher than 45 metres?
max of 50 and min of 4 ---> ampl = (50-4)/2 = 23
period is 30 min ---> 2π/k = 30 or k = π/15
You have:
a. If a child gets on the Ferris wheel when ,
how high will he be after riding for 15 minutes?
seems to be something missing after "wheel when, ..."
but after 15 minutes, which is half a period, the child would be at the max of 50 m
so far we have height = 23sin (π/15(t + d)) + 23
Just looking at my sketch, I would move the curve horizontally to the right 1/4 of a period so
that the height is zero when time is zero
so height = 23sin (π/15(t - 7.5)) + 23
let's see if that works
when t = 0, we want height = 0
0 = 23sin (π/15(0 + d)) + 23
sin (π/15(0 + d)) = -23/23 = -1
I know sin(3π/2) = -1 and also sin(-π/2) = -1
π/15(0 + d) = -π/2
π/15 d = -π/2
d = -7.5 , YES
https://www.wolframalpha.com/input/?i=y+%3D+23sin%28%CF%80%2F15%28x+-+7.5%29%29+%2B+23+from+0+to+32
looks like we got it.
height = 23sin (π/15(t - 7.5)) + 23
c) when t = 5
evaluate height, be replacing t with 5, watch the order of operation
looking at the graph, you should get about 11 m
d) set height = 45 and solve, you will get two answers ....
23sin (π/15(t - 7.5)) + 23 = 45
sin (π/15(t - 7.5)) = .956521.... , I stored it in calculator's memory
π/15(t - 7.5) = 1.2748... or π/15(t - 7.5) = π-1.2748... = 1.8667...
t = 13.586.. or t = 16.413..
So the child is higher than 45 m for about 16.412 - 13.586 or appr 2.8 minutes
check my arithmetic, this was fun!
a. To find the height after riding for 15 minutes, we need to determine the position of the Ferris wheel at that time.
Since it takes 30 minutes for one full rotation, after 15 minutes the wheel has completed half a rotation.
In half a rotation, the wheel goes from its maximum height to its minimum height and back to its maximum height.
Therefore, after riding for 15 minutes, the child will be at the maximum height of 50 meters.
b. To write a sine function to model the height of the child after boarding the Ferris wheel, we can consider the height as a function of time. Let's call the height h and the time t. The maximum height is 50 meters, the minimum height is 4 meters, and it takes 30 minutes for one full rotation.
A sine function that models the height of the child can be written as:
h(t) = (A/2)sin((2π/T)t) + (A/2) + C
In this equation:
A = amplitude = (maximum height - minimum height) / 2 = (50 - 4) / 2 = 23 meters (since it goes from maximum to minimum and back to maximum)
T = period = time for one full rotation = 30 minutes
C = vertical shift = average of the maximum and minimum heights = (50 + 4) / 2 = 27 meters
So the sine function to model the height of the child minutes after boarding the Ferris wheel is:
h(t) = 23sin((2π/30)t) + 27
c. To find the height of the child after riding for 5 minutes, we substitute t = 5 into the equation from part b:
h(5) = 23sin((2π/30)(5)) + 27
Calculating this, we get:
h(5) = 23sin(π/3) + 27
h(5) ≈ 23(0.866) + 27
h(5) ≈ 19.918 + 27
h(5) ≈ 46.918 meters
Therefore, after riding for 5 minutes, the child will be approximately 46.918 meters high.
d. To find the length of time the child will be higher than 45 meters, we need to consider the height function from part b:
h(t) = 23sin((2π/30)t) + 27
We set this equation greater than 45 and solve for t:
23sin((2π/30)t) + 27 > 45
Subtracting 27 from both sides, we get:
23sin((2π/30)t) > 18
Dividing by 23, we have:
sin((2π/30)t) > 18/23
To find the values of t that satisfy this inequality, we need to find the inverse sine of 18/23.
Using a calculator, we find that sin^(-1)(18/23) ≈ 0.789 radians.
So we have:
(2π/30)t > 0.789
Simplifying, we get:
t > (0.789*30)/(2π)
t > 3.798 minutes
Therefore, the child will be higher than 45 meters for any length of time greater than 3.798 minutes.
a. To determine how high the child will be after riding for 15 minutes, you can use proportionality. Since the Ferris wheel takes 30 minutes to make one full rotation, after 15 minutes, half of the rotation would have been completed.
To calculate the proportion of the height based on time, you can divide the time passed (15 minutes) by the total time for one rotation (30 minutes) and multiply it by the difference between the maximum height (50 meters) and the minimum height (4 meters).
So, the height after riding for 15 minutes would be:
(15 minutes / 30 minutes) * (50 meters - 4 meters) + minimum height
b. To write a sine function to model the height of the child minutes after boarding the Ferris wheel, you can use the concept of periodic motion and sine waves.
The general form of a sine function is: y = A * sin(B(x - C)) + D
Where:
- A represents the amplitude (half the difference between the maximum and minimum heights)
- B represents the frequency (the number of cycles or rotations within a specific time period)
- C represents the phase shift (the horizontal translation or where the graph starts)
- D represents the vertical shift (the minimum height)
In this case, the amplitude (A) would be (50 meters - 4 meters) / 2 = 23 meters.
The frequency (B) can be determined by the number of cycles in the total time of one rotation, which is 2π radians.
So the frequency (B) would be 2π / 30 (minutes) ≈ 0.2094 (rad/min).
The phase shift (C) would be 0 since we start measuring at the beginning.
And the vertical shift (D) would be the minimum height of 4 meters.
Therefore, the equation to model the height of the child would be:
y = 23 * sin(0.2094 * x) + 4
c. To determine how high is the child if they have been riding for 5 minutes, you can substitute the value of x as 5 in the sine function:
y = 23 * sin(0.2094 * 5) + 4
d. To find the time length during which the child will be higher than 45 meters, you can set the sine function greater than 45 and solve for x:
23 * sin(0.2094 * x) + 4 > 45
Then, solve the inequality to find the values of x that satisfy the condition.