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The vertical position, h, in metres, of a gondola on a Ferris wheel is modelled using the function

h = 10 sin ((pi/15) (t-d)) + 12

where t is the time, in seconds. The gondola must assume a vertical position of 7 m at t=0 s , and be on its way upward.

Determine the phase shift required for the desired operation of the Ferris wheel.

So we want h=7 when t=0 or

7 = 10sin((pi/15)(-d)) + 12
-.5 = sin((pi/15)(-d))
the reference angle is pi/6 .... ( sinpi/6 = 0.5)

so (-pi/15)d = pi + pi/6 or 2pi - pi/6 or -pi/6

if we let (-pi/15)d = -pi/6
d = 5/2

so h = 10 sin ((pi/15) (t-5/2)) + 12

check:
when t = 0, h = 10sin((pi/15)(-5/2) + 12
= 7
t = .01
h = 7.018 , a bit higher , so it is on its way upwards.

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