A male worker starts his shift at $8$:$00$ a$.$m$.$ He works at a machine for $110$ min during the morning, takes a $20-$min break before working another $110$ min, and then breaks for lunch. The lunch period is $30$ min. His afternoon routine is the same. He works for $110$ min, takes a $20-$min break, and then completes his shift working until $4$:$30$ p$.$m$.$ During the work periods, his energy expenditure rate is $6\mathrm{.}5$ kcal$/$min$.$ During the rest breaks, his energy expenditure rate is $1\mathrm{.}5$ kcal$/$min$.($a$)$ Compute the time$-$weighted average energy expenditure rate. $($b$)$ Using a maximum time$-$weighted average energy expenditure rate of $5\mathrm{.}0$ kcal$/$min as the standard or recommended level, what is the appropriate amount of rest time during the $4-$hour morning and afternoon work periods?