Answer: f(x) = 10in*cos(x*2*pi/60s) + 10in*sin(x*2*pi/60s) + 108in.
Step-by-step explanation:
When we have a circular motion, we can write this as:
f(x) = A*sin(k*x + f) + A*cos(k*x + f) + C.
where:
k is a constant that depends on the period.
f is a phase shift.
C is the mean height of the circular motion, in this case will be equal to the height of the center, that is 9ft.
C = 9ft.
A is the amplitude of the motion, in this case is equal to the radius, that is equal to the lenght of the second hand
A = 10in
But we should write both quantities in the same units, so we have:
1ft = 12in
then:
C = 9ft = 9*12in = 108in
Then our equation is:
f(x) = 10in*cos(k*x + f) + 10in*sin(k*x + f) + 108in
now, x = 0s coincides with the noon
so at x = 0s, the second hand is pointint to the twelve, this means that is in the max height, so we have:
F(0s) = 10in*cos(0 + f) + 10in*sin(0 + f) + 108 in
Now, the maximum value will be when the value inside the sinusoidals is 0.
(because cos(0) = 1 and sin(0) = 0)
then we have that f = 0.
Now, we know that it takes 60 seconds to do a full revolution, and for the sinusodial functions the period is 2*pi
Then we have:
c*60s = 2*pi
c = 2*pi/60s.
Then the equation is:
f(x) = 10in*cos(x*2*pi/60s) + 10in*sin(x*2*pi/60s) + 108in.
