Required information

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Taylor's is a popular restaurant that offers customers a large dining room and comfoltable bar area. Taylor Henry, the

owner and manager of the restaurant, has seen the number of patrons increase steadily over the last two years and is

considering whether and when she will have to expand its available capacity. The restaurant occupies a large home, and

all the space in the building is now used for dining, the bar, and kitchen, but space is available on the property to expand

the restaurant. The restaurant is open from $6$ p$.$m$.$ to $10$ p$.$m$.$ each night $($except Monday$)$ and, on average, has $28$

customers enter the bar and $54$ enter the dining room at the beginning of each of those hours. Taylor has noticed the

trends over the last $2$ years and expects that within about $4$ years, the number of bar customers will increase by $50\%$ and

the dining customers will increase by $20\%.$ Taylor is worried that the restaurant will be not be able to handle the increase

and has asked you to study its capacity. In your study, you consider four areas of capacity: the parking lot $($which has $84$

spaces$),$ the bar $(58$ seats$),$ the dining room $(104$ seats$),$ and the kitchen. The kitchen is well$-$staffed and can prepare any

meal on the menu in an average of $12$ minutes per meal. The kitchen, when fully staffed, is able to have up to $20$ meals in

preparation at a time, or $100$ meals per hour $(60$ minute $12$ minute $20$ meals$).$ To assess the capacity of the restaurant,

you obtain the additional information:

Diners typically come to the restaurant by car, with an average of $3$ persons per car, while bar patrons arrive with an

average of $1\mathrm{.}5$ persons per car.

Diners, on average, occupy a table for an hour, while bar customers usually stay for an average of $2$ hours.

Due to fire regulations, all bar customers must be seated.

The bar customer typically orders one drink per hour at an average of $$8$ per drink; the dining room customer orders a

meal with an average price of $$21$; the restaurant's cost per drink is $$2,$ and the direct costs for meal preparation are $$4.$

Required:

$1-$a$.$ Given the current number of customers per hour, what is the amount of excess $c$

kitchen?