$(3$ points$)$ A new project involves the construction of columns in a building. A total of $30$ different columns need to be constructed. It takes one worker an average of $14$ hours to construct one column. Also, assume $8-$hour workdays.

a$.$ If $3$ workers are assigned to the task of constructing the $30$ columns, determine the productivity $($columns$/$day$)$ for all workers working together. Assume the productivity rate for one worker when they work alone is the same as when they work with the other workers.

b$.$ What duration $($nearest whole day$)$ should be assigned to the activity to construct the columns for all $30$ columns using $3$ workers?

c$.$ How many more workers are needed to decrease the activity duration calculated in Part $($b$)$ by $30\%?$ Use the actual duration calculated in Part $($b$),$ not the duration rounded to the nearest whole day.

$(3$ points$)$ An engineer predicts that designing the formwork for $32$ precast concrete girders will require an average of $120$ hours. The time needed fluctuates due to variations in girder designs and differences in the engineers' productivity. The durations follow a normal distribution with a standard deviation of $1\mathrm{.}84$ hours.

a$.$ What is the probability that the engineer will complete the design of the formwork for all of the girders in $124$ hours or less? $(P_{124})$

b$.$ What is the probability that the designs will take no more than $118$ hours? $(P_{118})$

c$.$ When planning the design activity, what activity duration $($in hours$)$ should be used to provide an $85\%$ level of confidence that the duration will not be exceeded?

$(3$ points$)$ Based on historical project data, a contractor estimates that it typically takes an average of $20\mathrm{.}0$ days to complete the formwork for $30$ concrete walls, with a standard deviation of $1\mathrm{.}55$ days. This variability is attributed to factors such as weather conditions, worker performance, and sitespecific impacts. It is assumed that the activity durations follow a standard normal distribution.

a$.$ What is the probability that the duration to construct the formwork for all of the walls will take longer than $22$ days?

b$.$ After the formwork is constructed, the wall reinforcing will be installed and then the concrete placed. These three activities will be performed in series $($i$.$e$.,$ finish$-$to$-$start relationships$).$ For this part, use the following values:

Formwork construction:

Rebar installation:

$=20\mathrm{.}0$ days, $=1\mathrm{.}55$ days

$=7\mathrm{.}25$ days, $=0\mathrm{.}50$ days

$=9\mathrm{.}50$ days, $=0\mathrm{.}70$ days

Place and cure concrete:

What is the probability that all three activities will be completed in a total of $35$ days or less? Hint: Use the mean, standard deviation, and variance for the entire path $($similar to the process used for PERT$).$