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Rectangle Partitioned by Curves

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August 9, 2024
geometryrectangleoptimizationcalculus2022

Problem Statement

Consider the rectangle in the coordinate plane with corners (0,0)(0, 0), (16,0)(16, 0), (16,4)(16, 4), and (0,4)(0, 4). For a constant x0∈[0,16]x_0 \in [0, 16], the curves \{(x, y) : y = \sqrt{x} \,\text{ and }\, 0 \leq x \leq 16\}   \text{and}   \{(x_0, y) : 0 \leq y \leq 4\} partition this rectangle into four 2D regions. Over all choices of x0x_0, determine the smallest possible sum of the areas of the bottom-left and top-right 2D regions in this partition. (The bottom-left region is {(x,y):0≤x<x0  and  0≤y<x}\{(x, y) : 0 \leq x < x_0 \,\text{ and }\, 0 \leq y < \sqrt{x}\}, and the top-right region is {(x,y):x0<x≤16  and  x<y≤4}\{(x, y) : x_0 < x \leq 16 \,\text{ and }\, \sqrt{x} < y \leq 4\}.)
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