1A research firm finds that the average number of customers that are in Somerset Mall on a

Saturday varies sinusoidally with time. The mall is open from 9:00 am to 9:00 pm; the minimum

number of customers is at 9:00 am and 9:00 pm when there are zero customers. The maximum

number of customers is at 3:00 pm when there are 875 customers in the mall,

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