The productivity of a company during the day is given by # Q(t) = -t^3 + 9t^2 +12t # at time t minutes after 8 o'clock in the morning. At what time is the company most productive?

Answer 1

2:36 pm

The productivity is given as:

# Q(t) = -t^3 + 9t^2 +12t #
To find the optimum productivity we seek a critical point of #Q(t)#, and would expect to find a maxima.
Differentiating wrt #t# gives:
# (dQ)/dt = -3t^2 + 18t +12 #
At a critical point #(dQ)/dt=0 => #
# -3t^2 + 18t^ +12 = 0 # # :. t^2 -6t^ -4 = 0 # # :. t=3+-sqrt(13) #
We require #t>0 => t=3+sqrt(13)#

We can do a second derivative test to verify this is a maximum;

# (d^2Q)/dt^2 = -6t + 18 #
When #t=3+sqrt(13) => (d^2Q)/dt^2 <0 => # maximum
Thus the maximum productivity occurs when #t=3+sqrt(13)#
ie, # t ~~ 6.60555 ... # which correspond to a duration of 6h36m
As #t=0# was t 8AM then the optimum production time would be 2:36 pm
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the time when the company is most productive, we need to find the maximum value of the function Q(t). To do this, we'll first take the derivative of Q(t) with respect to t and then set it equal to zero to find the critical points. Then we'll determine which critical point corresponds to a maximum value.

Q'(t) = dQ(t)/dt = -3t^2 + 18t + 12

Setting Q'(t) equal to zero:

-3t^2 + 18t + 12 = 0

Solving this quadratic equation will give us the critical points. Once we have the critical points, we'll plug them back into the original function Q(t) to find which one yields the maximum value, as the maximum productivity corresponds to the maximum value of Q(t).

After finding the value of t corresponding to the maximum productivity, we can convert it into hours and minutes to determine the time at which the company is most productive after 8 o'clock in the morning.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7