$1.$ A company manufactures purses, bags and backpacks. The manufacturing process requires

leather and synthetics, leather being the scarce raw material. The manufacturing process

requires two types of skilled labour $-$ sewing and finishing. The following table gives the

availability of the resources, their usage by the three products and the profits per unit.

Resource requirements per unit

Resource Purse Bag Backpack Daily availability

Leather $($square feet$)21342$ square feet

Sewing $($hours$)21240$ hours

Finishing $($hours$)10\mathrm{.}5145$ hours

Selling price $($dollars$)242245$

$($a$)$ Formulate a linear program to maximize the profit

$($b$)$ Find the optimal solution and the optimal profit using the simplex method.