4. A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy-plane bounded by the lines y = 0, y = 3x, and y = 30 – 2x. What is the largest possible area of such a rectangle?

Consider the following graph of the closed region described above with an inscribed rectangle of length L and width W = W1 = W2:

To represent L in terms of x, use the relation between W1 and W2, as follows,

So that the area A in terms of x is given by,

Hence, the largest possible area of such a rectangle is obtained by finding the maximum value of A(x). Which is the value of A(x) evaluated at the value of the solution of A’(x) = 0.