Answer is: option3

11:00 am

Solution:

We need to determine the time at which the rate of production, \( R(t) \), is increasing most rapidly. This means finding when \( R'(t) \) reaches its maximum.

From the given solution:

\[ R'(t) = 48 - 12(t - 3)^2 \]

To find the maximum of \( R'(t) \), we compute its derivative:

\[ R''(t) = -24(t - 3) \]

Setting \( R''(t) = 0 \):

\[ -24(t - 3) = 0 \] \[ t = 3 \]

This indicates that \( R'(t) \) reaches a maximum at \( t = 3 \).

Since \( t \) represents hours since 8:00 AM, we compute:

\[ 8:00 \text{ AM} + 3 \text{ hours} = 11:00 \text{ AM} \] Final Answer

The rate of production is increasing most rapidly at 11:00 AM.