the productivity of a person at work (on a scale of 0-10) is modeled by a cosine function: 5cos (pie/2t) +5, where t is in hours. If the person starts work at t=0, being 8:00 am, at what times is the worker the least productive?

a) 12pm
B) 10 am and 2pm
C) 11 am and 3 pm
D) 10 am,12pm and 2 pm

MULTIPLE CHOICE

cos has max at t=0

cos(pi/2 t) has period 4
cos has min at 1/2 the period.

so,
f(0) (8am) = 5*1+5 = 10
f(1) (9am) = 5*0+5 = 5
f(2) (10am) = 5*(-1) + 5 = 0
...
least productive at 10:00,14:00,...
(B)

Well, let's analyze the cosine function. As you might know, the cosine function oscillates between -1 and 1. In this case, we have 5cos(pi/2t) which means the maximum value will be 5 and the minimum value will be 0.

Since the equation is 5cos(pi/2t) + 5, we simply shift the graph up by 5 units. This means the maximum productivity will be 10 and the minimum will be 5.

So, the worker will be least productive when their productivity is 5.

Looking at the options:

a) 12 pm - This is not the least productive time since the worker would be halfway through their workday.
B) 10 am and 2 pm - These could be potential options since they fall just before and after the midpoint of the workday.
C) 11 am and 3 pm - These could also be potential options since they fall closer to the midpoint of the workday.
D) 10 am, 12 pm, and 2 pm - This option seems to include all the potential times.

Based on the analysis, the correct answer should be:

D) 10 am, 12 pm, and 2 pm

To determine the times when the worker is least productive, we need to find the values of t that minimize the cosine function.

Since the cosine function has a period of 2π, we can rewrite it as:

5cos(π/2 * t) + 5

To find the minimum values of this function, we need to find the values of t that make the cosine function equal to its minimum value, which is -1.

Setting the cosine function equal to -1, we have:

-1 = 5cos(π/2 * t) + 5

Subtracting 5 from both sides, we get:

-6 = 5cos(π/2 * t)

Dividing both sides by 5, we have:

-6/5 = cos(π/2 * t)

To find the inverse cosine, we have:

π/2 * t = arccos(-6/5)

Solving for t, we get:

t = (2/arccos(-6/5)) * π

Using a calculator, we find that:

t ≈ 0.8899

Since t represents hours after 8:00 am, the time when the worker is least productive is approximately 8:53 am.

Therefore, the correct answer is (B) 10 am and 2 pm.

To find the times when the worker is the least productive, we need to identify the minimum values of the cosine function.

The general form of the cosine function is given by:

y = A*cos(Bx + C) + D

In this case, the function is:

y = 5cos(π/2t) + 5

So, A = 5, B = π/2, C = 0, and D = 5.

The cosine function has a period of 2π, meaning it repeats every 2π units. In this case, since t is in hours, the period will be 2π hours.

The minimum value of the cosine function occurs halfway through its period, which is at t = π/2B.

Substituting the given values, we have:

t = π/2(π/2) = 1 hour

Now, we need to convert this time into the corresponding actual time in the day.

The person starts work at t = 0, which is 8:00 am. Therefore, t = 1 hour corresponds to 8:00 am + 1 hour = 9:00 am.

So, the worker is the least productive at 9:00 am.

Therefore, the answer is:

C) 11 am and 3 pm