Answer:
x = 18 cm
y = 9 cm
Step-by-step explanation:
An open-top box will have minimum surface area when it is in the shape of half a cube. The whole cube would have volume 2×2916 = 5832 cm³, so its side length would be ∛5832 cm = 18 cm.
The dimensions of the box are x = 18 cm; y = 9 cm.
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The surface area is x^2 +4xy, where y = 2916/x^2. That is, the surface area is ...
S = x^2 +11664/x
Setting the derivative to zero, we find ...
dS/dx = 0 = 2x -11664/x^2
x^3 = 5832 . . . . . . . . may look familiar
x = ∛5832 = 18
y = 2916/18^2 = 9
