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