Answer: C(t) = -475.5*cos(t*pi/6) + 475.5
Step-by-step explanation:
We know that we have a sinusoidal relation, with a minimum at 9:00 am and at 9:00 pm.
If we define the 9:00 am as our t = 0, we have that the maximum, at 3:00pm, is at t = 6 hours.
and the other minimum, at 9:00pm, is at t = 12 hours.
Then we need to find a trigonometric function that has the minimum at t = 0, we can do this as:
-Cos(c*t)
when t = 0
-cos(0) = - 1
then we have a function:
C(t) = -A*cos(c*t) + B
where A, c and B are constants.
We know that at t = 0 we have 0 customers, and that at t = 6h we have 875 customers, that is the maximum.
then:
C(0) = 0 = -A + B
this means that A = B, then our function is:
C(0) = -A*cos(c*t) + A.
now, at t = 6h we have a maximum, this means that -A*cos(c*6h) = A
then:
C(6h) = A + A = 875
2A = 875
A = 875/2 = 437.5
and we also have that, if -cos(c*6) = 1
then cos(c*6) = -1
and we know that cos(pi) = -1
then c*6 = pi
c = pi/6
Then our function is
C(t) = -475.5*cos(t*pi/6) + 475.5
