The height h in feet above ground of a seat on a Ferris whee

The height h (in feet) above ground of a seat on a Ferris wheel at time t (in minutes) can be modeled by h(t) = 53 + 50 sin(pi/18t - pi/2.). The wheel makes one revolution every 36 seconds. The ride begins when t = 0. During the first 36 seconds of the ride, when will a person on the Ferris wheel be 53 feet above ground? (Enter your answers as a comma-separated list.) When will a person be at the top of the Ferris wheel for the first time during the ride? t = s If the ride lasts 180 seconds, then how many times will a person be at the top of the ride? times At what times? (Enter your answers as a comma-separated list.) t = s

Solution

Solution:

a) The person will be 53 ft above ground when h(t) = 53, and this happens when

sin(pi/18 t - pi/2) = 0 , hence pi/18 t - pi/2 = k pi ==> t/18 = k + 1/2 ==> t = 18 k + 9

The possible values if 0 < t < 36 are: 9 and 27 seconds

b) The person at the top at t = 18. To see the times the person will be on top, we have to
solve the equation sin(pi/18 t - pi/2 ) = 1 ==> pi/18 t - pi/2 = pi/2 + 2k pi

Multiplying by (18/pi) gives t = 18 + 36 k, where k = 0, 1, 2, ... etc.

The values are 18, 54, 90, 126, 162.

So the person will be at the top 5 times. Note that we could have calculated the no. of times by
solving for k such that 18 + 36 k <= 180, from which 36 k <= 162 ==> k <= 162/36 = 9/2 = 4.5

From which no. of times is 4 + 1 = 5 times.