The personnel manager for a construction company keeps track of the total number of labor hours spent on a construction job each week during the construction. Some of the weeks and the corresponding labor hours are given in the table.

Cumulative Labor$-$Hours by the Number of Weeks after Job Begins

$\backslash$table$[[$Weeks $($x$),$Hours $(f)],[1,24],[4,160],[7,1251],[10,5632],[13,9278],[16,10,011],[19,10,098]]$

$($a$)$ Find the function for the logistic model that gives total number of labor hours where $x$ is the number of weeks after construction begins, with data from $1x19.($Round all numerical values to three decimal places.$)$

$f(x)=$

$($b$)$ Write the derivative equation for the model. $($Round all numerical values to three decimal places.$)$

$f^{\text{'}}(x)=$

$($c$)$ On the interval from week $1$ through week $19,$ when is the cumulative number of labor hours increasing most rapidly? $($Round your answer to three decimal places.$)$ weeks

How many labor hours are needed in that week? $($Round your answer to three decimal places.$)$

$$ labor hours

$($d$)$ If the company has a second job requiring the same amount of time and the same number of labor hours, a good manager will schedule the second job to begin when the number of cumulative labor hours per week for the first job begins to increase less rapidly. How many weeks into the first job should the second job begin? weeks