Question $3.$ A cargo plane has three compartments for storing cargo: front, center and back. These compartments have capacity limits on both weight and space, as summarized below:

$\backslash$table$[[$Compartment$,\backslash$table$[[$Weight Capacity$],[($Tons$)]],\backslash$table$[[$Space Capacity$],[($cubic feet$)]]],[$Front$,12,7000],[$Center$,18,9000],[$Back$,10,5000]]$

The following four cargoes have been offered for shipment on an upcoming flight as space is available:

$\backslash$table$[[$Cargo$,\backslash$table$[[$Weight$],[($tons$)]],\backslash$table$[[$Volume$],[($cubic feet$/$tons$)]],\backslash$table$[[$Profit$],[($$$/$tons$)]]],[1,20,500,320],[2,16,700,400],[3,25,600,360],[4,13,400,290]]$

Any portion of these cargoes can be accepted.

The objective is to determine how much $($if any$)$ of each cargo should be accepted and how to distribute each among the compartments to maximize the total profit for the flight.

a$.$ Formulate a linear programming model for this problem. $($no need to solve with a software$)$