the productivity of a person is modelled by the function y=5cos(pi/2 t)+5, where t is in hours and t>and equal to 0. if the person starts work at 8:00am and works for 12 hours determine at what times they are least productive.assume that t=0 represents 8:00am.
Reiny did this one already. See the related questions below.
To determine the times when the person is least productive, we need to find the minimum points of the function.
The given function that models productivity is: y = 5cos(π/2 t) + 5
To find the minimum points, we need to find where the derivative of the function is equal to zero.
First, let's find the derivative of the function:
y' = d/dt (5cos(π/2 t) + 5)
Using the chain rule, we can differentiate the function as follows:
y' = -5(π/2)sin(π/2 t)
Setting y' = 0 to find the times of minimum productivity:
-5(π/2)sin(π/2 t) = 0
sin(π/2 t) = 0
To find the values of t that satisfy sin(π/2 t) = 0, we need to determine the values of t for which sin(x) = 0. We know that sin(x) = 0 when x is an integer multiple of π.
So, we have:
π/2 t = nπ
where n is an integer.
Solving for t, we get:
t = 2n, where n is an integer
Now, let's consider the given conditions: the person starts work at 8:00 am and works for 12 hours.
If t represents the number of hours since 8:00 am, then we want to find the values of t within the range 0 ≤ t ≤ 12.
The values of t that satisfy the conditions are:
t = 2(0) = 0 (8:00 am)
t = 2(1) = 2 (10:00 am)
t = 2(2) = 4 (12:00 pm)
t = 2(3) = 6 (2:00 pm)
t = 2(4) = 8 (4:00 pm)
t = 2(5) = 10 (6:00 pm)
t = 2(6) = 12 (8:00 pm)
Therefore, the person is least productive at 10:00 am, 12:00 pm, 2:00 pm, 4:00 pm, 6:00 pm, and 8:00 pm.