$1.$ What is the cycle time for this line?

The cycle time is $1$ minute for this line. This is because cycle time is defined as the time between units being completed and each line station$$s duration is a minute, making the time between units being completed equal to a minute.

$2.$ What is the line$$s output $($R$)$ during a $480-$minute $($eight$-$hour$)$ period?

The line$$s output during a $480-$minute period is $480$ units. This is because the cycle time is a minute, and if you have $480$ minutes you should be able to produce $480$ units in that time.

$3.$ What is the utilization of each workstation during the $480-$minute period?

The utilization is $100\%$ because the time spent working at each station is equal to the cycle time.

$4.$ If Workstation $2$ had a processing time of $2$ minutes, what is the cycle time, output in $480$ minutes, and the utilization of each workstation?

The cycle time would then be two minutes and the output would be $240$ units. The utilization of workstation $1$ would still be $100\%,$ but the utilization of workstation $2$ would go down to $66\mathrm{.}67\%$

$5.$ If you invested in improving the process, where would you make changes? Why?

Since workstation $2$ would be the bottleneck, I would make changes to adjusting the processing time.