Problems at the carnival

Question

A Ferris wheel is 35 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 6 minutes. How many minutes of the ride are spent higher than 32 meters above the ground? Give your answer correct to the nearest second.

I was trying to help my daughter solve this for her college assignment. When she puts in the answer on the online assignment, it tells her the answer of 1 minute is correct, but we can't get the seconds correct. Can someone please assist with the correct way to solve this?

Answer 1

period = 6

B= 2pi/period = 2pi/6 = pi/3

period = 2pi/|B|

construct a

sinusoidal function or curve equation

= Asin[B(t-C)] + D

where

A= amplitude= half the range= radius

C= phase shift or horizontal shift

D = midline or vertical shift=minimum + radius

35 feet diameter, 2 feet lowest point

radius = 35/2 = 17.5. midline = 19.5= D

amplitude = 17.5= A

f(t) =17.5sin[A(t-C)]+D

f(t) = 17.5sin[(pi/3)(t-C)]+ 19.5

6 o clock is when level = 2 feet and t=0

sin[(pi/3)(C)] =(2-19.5)/17.5 = -1

sin(piC/3)=-1

(pi/3)(C) = 3pi/2 or -pi/2

C = (-pi/2)(3/pi)

C=-3/2

f(t) = 17.5sin[(pi/3)(t+3/2)] +19.5

f(t) = 32= 17.5sin[(pi/3)(t+3/2)] + 19.5

solve for t

then find the difference in t from time at the top of the ferris wheel, at t=+/-3, +/-9, ...

the multiply that by 2

the calculations get tedious

it seems to come out to 1 minute and 28.8 seconds, rounded off

another way to look at it is draw a diagram.

find the angle formed from the center of the ferris wheel to the bottom

and to the point on the wheel where height =32 feet, from f(0) = 2 feet to f(t)=32 feet

sine of the angle formed with the horizontal is 5/7

the angle formed with the vertical is sin^-1(5/7) + 90 = 135.585 degrees

(135.585/360)(6) =2.25975

2(3-2.25975) = 2(.74025) = 1.4805 minutes

= 1 minute and .4805/60 seconds

= 1 minute and 28.83 seconds

= 1 minute and 29 seconds

if you work a problem 2 different ways and get the same answer, that tends to suggest it's correct

you could also have used a sinusoidal function with cosine instead of sine

Answer 2

I get 1.9665 minutes

0.9665(60) = 58

1 minute 58 seconds