$4)($extra credit$)$ A greengrocer has exactly $1,000$ square feet available to display and sells $3$ kinds of vegetables. The space consumed by each kind of vegetable is proportional to its cost, and tomatoes consume $0\mathrm{.}5$ square feet per pound. There is a $$100$ setup cost for replenishing any of the vegetables, and the interest rate is $25\%$ per annum. The $3$ vegetables for sale have the following data: $$ 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:

$4\mathrm{.}1)(40$ points$)$ What is the optimal quantity for purchasing each vegetable $($include all formulas$)?$