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If you have two pairs of numbers whose sum is the same, their product increases as the two numbers get nearer to half the sum. The product is a maximum at that value.

70 * 30 = (50+20)*(50-20)

The right hand side is the factorisation of the difference of two squares, and can be shown to be equal to

502 - 202

Since 202 must be positive (it is a square), 70*30 must be less.

For all values of x, (50+x)*(50-x) is less than or equal to 50*50=2500

with equality attained only when x = 0.

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Q: Why does the area of 70 times 30 equal 2100 but the area of 50 times 50 equal 2500 square feet?
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