2. A shoemaker has a supply of 100 square feet of type A leather which is used for soles and 600 square feet of type $B$ leather which is used for the rest of the shoe. The average pair of shoes use $\frac{1}{2}$ square feet of type $A$ leather and 2 square foot of type B leather. The average pair of boots use $\frac{1}{2}$ square feet of type A leather and 6 square feet of type $B$ leather. If shoes sell for $\$ 40$ a pair and boots sell for $\$ 60 \mathrm{a}$ pair, find the maximum income.

Step 1 ：Let's denote the number of pairs of shoes by x and the number of pairs of boots by y. The objective function to maximize is \(40x + 60y\) (the total income from selling shoes and boots).

Step 2 ：The constraints are: The total amount of type A leather used cannot exceed 100 square feet: \(0.5x + 0.5y \leq 100\). The total amount of type B leather used cannot exceed 600 square feet: \(2x + 6y \leq 600\).

Step 3 ：Solving this problem using a linear programming solver, we get the optimal values of x and y.

Step 4 ：The maximum income the shoemaker can earn is $9000 by making 150 pairs of shoes and 50 pairs of boots. This is the optimal solution given the constraints of the available leather.

Step 5 ：Final Answer: The maximum income is \(\boxed{9000}\).