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 \(\boxed{9000}\).