please show all work hand-written

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Two adjacent rectangular gardens will be built using 2000 feet of fencing material; the goal is to cover the largest possible area. Let x be the length of the shared side. Write down the objective function in terms of x. Find the largest possible area. An open-top box will be made from a rectangular cardboard measuring 40m by 60in. by cutting congruent square corners and folding the sides. Let x denote the side length of each square corner to be cut. The goal is to maximize the volume. Write down the objective function in terms of x. Let m and n to positive real numbers. Given 8m + 15n = 40, find the maximum value for the product mn.. A rectangle is drawn in the rst quadrant no that the base lies on the x-axis, bottom left corner on the origin, top right corner is on the line 2y + 4x : 40 . Draw a diagram. Find the maximum area of such a rectangle. . A rectangle is drawn in the rst and second quadrants so that the base lies on the x-axis, upper left corner on the line y = 2x + 6 , top right corner is on the line y = 2x + 6. Draw a diagram. Find the maximum area of such a rectangle. A rectangle is drawn so that the base lies on the x-axis, and the top corners are on the curve y =48 - x . Draw a diagram. Find the maximum area of such a rectangle