Question

Optimization. A rectangle is to have an area of 32 square cms. Find its dimensions so that

Answer 1

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xy = 32 makes y = 32/x z² = (32/x)² + (x/2)²

Isolate z
z = 32/x + x/2 

Find dz/dx

fr(z) = (x² - 64)/ (2x²)

Set z' = 0, algebraically solving for x having
x +- 18

Since you cannot have a dimension of -8, the only answer for x is 8.

Go back to the original area equation; 32 = xy and plug in x making
32 = 8y
y = 4