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.
To determine the phase shift, we need to find the value of 'd' in the equation:
h = 10 sin ((π/15) (t - d)) + 12
We are given that the gondola must assume a vertical position of 7 m at t = 0 s and be on its way upward. This means that when t = 0, h = 7.
Substituting the values into the equation:
7 = 10 sin ((π/15) (0 - d)) + 12
Rearranging the equation, we get:
-5 = 10 sin (-πd/15)
Dividing by 10, we have:
-0.5 = sin (-πd/15)
To find the phase shift 'd', we need to invert the sine function and solve for 'd':
sin^-1(-0.5) = -πd/15
Since sin^-1(-0.5) = -π/6, we have:
-π/6 = -πd/15
Cross multiplying, we get:
-15*π/6 = -πd
Canceling out 'π', we have:
-15/6 = -d
Simplifying, we find:
d = 15/6 = 2.5
Therefore, the phase shift required for the desired operation of the Ferris wheel is 'd = 2.5'.
To determine the phase shift required for the desired operation of the Ferris wheel, we need to find the value of 'd' in the equation h = 10 sin ((π/15) (t-d)) + 12.
Given that the gondola must assume a vertical position of 7 m at t=0 s and be on its way upward, we can substitute these values into the equation and solve for 'd'.
Substituting t=0 and h=7 into the equation, we have:
7 = 10 sin ((π/15) (0-d)) + 12
Simplifying this equation:
7 - 12 = -10 sin ((π/15) (-d))
-5 = -10 sin ((π/15) (-d))
Dividing both sides of the equation by -10:
0.5 = sin ((π/15) (-d))
To determine the value of (-d) that satisfies this equation, we need to find the inverse sine (or arcsine) of 0.5. Using a calculator, we find that the inverse sine of 0.5 is 30 degrees (or π/6 radians).
So, we have:
(π/15) (-d) = π/6
Rearranging the equation:
(-d) = (π/6)(15/π)
Simplifying:
(-d) = 5/2
Finally, multiplying both sides of the equation by -1:
d = -5/2
Therefore, the phase shift required for the desired operation of the Ferris wheel is d = -5/2.