$32$ A tourist group wants to stay over for one day at the Eiland Spa Resort and must book

their accommodation in advance. The resort has a choice of two types of chalets

$($cottages$).$

Type A: These chalets can host a couple and has a double bed.

Type B: These chalets can host a family and has a double bed and two single beds.

In the group there are at least $4$ couples who want to stay in the type A chalets and

they want not less than $4$ of the type $B$ chalets to place families with four members.

The ratio of type A to type B must be at least one to two. The group needs $18$ chalets

at the most.

Let the number of type A chalets be $x$ and the number of type B be $y.$

$3\mathrm{.}2\mathrm{.}1$ Two of the constraints are $x0$ and $y0.$

Model the other constraints in respect of the above information in terms

of $x$ and $y.$

$3\mathrm{.}2\mathrm{.}2$ The tariff for type A is R$1000/$day and type B is R$1500/$day$.$

Write the equation that represents the total cost for the stay in terms of

$x$ and $y.$

$3\mathrm{.}2\mathrm{.}3$ Determine the gradient for the search line. Show the steps.

$3\mathrm{.}2\mathrm{.}4($a$)$ Using a scale of $1$ unit $=2$ chalets, represent the inequalities

$($b$)$ Indicate the feasible region on the graph.

$($c$)$ Draw the search line on the graph and calculate the maximum cost

for theirstay.

$3\mathrm{.}2\mathrm{.}5$ Due to cancellations of tourists, the group on arrival consisted of $10$

families who needed type B chalets and $3$ couples who required type $A$

chalets.

Will the tour leader be able to meet the requirements of all the tourist in

their booked chalets? Give a reason for your answer with reference to the

feasible region.