$4.$ A green grocer has exactly $1,000$ square feet $($sqf$)$ available to display and sell $3$ kinds of vegetables. The ratios of each vegetable space$-$per$-$pound to cost$-$per$-$pound are the same for all vegetables, and tomatoes consume $0\mathrm{.}5$ sqf per $.$ There is a $$100$ setup cost for replenishing any of the, and the interest rate is $25\%$ per annum. The $3$ vegetables for sale have the following data:

Tomatoes: annual demand is $850$ pounds and the cost per pound is $$0\mathrm{.}29.$

Lettuce: annual demand is $1280$ pounds and the cost per pound is $$0\mathrm{.}45.$

Zucchini: annual demand is $630$ pounds and the cost per pound is $$0\mathrm{.}25.$

Answer the following questions:

$4\mathrm{.}1)$ What is the optimal quantity for purchasing each vegetable $($include model name and all formulas$)?$