QUESTION ONE $[2$O MARKS$]$

A street vendor in Nairobi City sells hot dogs and hamburgers. A hot dog gives a profit of $Sh\mathrm{.}0\mathrm{.}60$ and a hamburger gives a profit of Sh$.0\mathrm{.}50.$ Each hot dog uses Sh$.1\mathrm{.}20$ worth of materials and each hamburger uses Sh$.1\mathrm{.}60$ worth. There is only Sh$.48$ worth of materials currently on hand to make hot dogs and hamburgers. From previous experience the vendor knows that he should make at least $8$ hamburgers and no more than $20$ hot dogs. He also believes that he should make at most twice as many hot dogs as hamburgers.

a$.$ Solve the problem graphically, stating optimal values of the decision variables. What is the maximum profit?

$[15$ Marks$]$

b$.$ How low would the profit for hot dogs have to drop before the vendor would stop making them?

$[2$ Marks$]$

c$.$ The vendor remembers that he omitted the constraint that no more than $12$ hamburgers should be produced, all other constraints remaining the same. With this new constraint in the model, what would the optimal solution be$?$ What profit would the vendor make now? $[3$ Marks$]$