Mosaic Tile Company plans to produce two basic types of tile for home use: a larger single$-$ colored tile and a smaller patterned tile. The tiles are produced in batches of $100.$ The first step is to pour the clay into molds. It takes $27$ minutes $(\mathrm{.}45$ hours$)$ to mold a batch of larger tiles and $18$ minutes $(\mathrm{.}30$ hours$)$ to mold a batch of smaller tiles. The company has $75$ hours available each week for molding. After the tiles are molded, they are baked in a kiln $--\mathrm{.}40$ hours for $100$ larger tiles and $\mathrm{.}60$ hours for a batch of $100$ smaller tiles. The company has $105$ hours available each week for baking. After baking, the tiles are glazed, and this process takes $\mathrm{.}15$ hours for a batch of $100$ larger tiles, and $\mathrm{.}25$ hours for a batch of $100$ smaller tiles. $60$ hours are available each week for the glazing process. Each batch of $100$ large tiles requires $30$ pounds of clay to produce, while each batch of smaller tiles requires $20$ pounds of clay. The company has $6000$ pounds of clay available each week. Mosaic Tile earns a profit of $$190$ for each batch of $100$ larger tiles and $$240$ for each batch of $100$ smaller tiles. $7$a$)$ How many batches of each type of tile should be produced to maximize profit? $7$b$)$ What is the maximum weekly profit the company can earn? $7$c$)$ How many hours of baking resources are left over$/$not used at the optimal solution point? $7$d$)$ How much would an additional hour of molding add to the profit? $7$e$)$ For artistic reasons, Mosaic's owner likes to produce the smaller tiles best. She also believes in the long run that the smaller tiles will be a more successful product. What must be the profit for the smaller tiles in order for the company to produce only smaller tiles? $7$f$)$ The company that supplies Mosaic with clay has indicated that it can deliver an additional $100$ pounds of clay each week. Should Mosaic agree to this offer?