A person's trip on a carnival ferris wheel is modelled by the equation: y=7sin(pi(t-4)/8) + 9.2 How high above the ground is the bottom of the wheel?

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thilina-g | College Teacher | (Level 1) Educator

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Assuming y = 0 is the ground level.

`y = 7sin(pi/8(t-4))+9.2`

The bottom of the wheel is the minimum point of this man's trip. So if we differentiate y wrt t, we will be able to find the minimum point of the the trip.

`(dy)/(dt) = 7*pi/8*cos(pi/8(t-4))=(7pi)/8cos(pi/8(t-4))`

for extreme points, the derivative must be zero. Therefore,

`(7pi)/8cos(pi/8(t-4)) = 0`

`cos(pi/8(t-4)) = 0`

let's say `pi/8(t-4) = alpha`

`cos(alpha) =0` this gives the primary solution as `alpha = pi/2`

` `


Now we have to find the general solution of `alpha`

The general solution for cosine is,

`alpha = 2npi+-pi/2` where n is any integer.

n =0,

`alpha = -pi/2 or pi/2`

n =1,

`alpha = (3pi)/2 or (5pi)/2`

we have a range of solutions, since this is a sinusoidal function. The maximums and minimums are repeating. So it's enough for solutions (we don't need to find all the solutions)

now `y = 7sin(alpha)+9.2`

if we find the second derivative,

`(d^2y)/(dt^2) = -(7pi^2)/64sin(pi/8(t-4)) = -(7pi^2)/64sin(alpha)`

If we calculate the sign of the second derivative at our solutions,

at `alpha = -pi/2`   ,  `(d^2y)/(dt^2) =+positive`

Therefore at alpha = -pi/2,  The first minimum exists,


`y_min = 7sin(-pi/2)+9.2 = -7+9.2 = 2.2 m `


The height to the bottom of the wheel is 2.2 meters.


Actually there is a short way to find this solution,

`y = 7sin(alpha)+9.2`

The lowest value sin(alpha) can get is (-1). Therefore the lowest value y can get is,

y_min = 7*(-1)+9.2 = 2.2 m. (same answer)