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For the first problem, cot(A) = 8/6, so, using trigonometry, you can find sin(A) and express A'A'' through x. Triangle ABC is similar to A'B'C, so B'C = 4x/3, which lets you find A'B'. Now you can express the area of the grey rectangle through x. Note that A'A'' is proportional to 6 - x and A'B' is proportional to x, so the area is proportional to x(6 - x). This means the maximum is achieved when x = 3. To calculate the exact area, you need to find the coefficients of proportionality.