With a given symmetric surface area, the maximum volume is achieved with a symmetric shape. In this case, the rectangular solid would become a cube, such that:
6x²=289
x=sqrt(289/6)
volume = x^3 = (289/6)^(3/2)=334.3
and the side = x = 6.94 metres
Using calculus:
If the side of the base is x,
Height, h = (289-2x²)/4x
Volume,
V = x²h
=x²((289-2x²)/(4x))
=289x/4 - x³/2
Calculate dV/dx and equate to zero.
dV/dx = 289/x - 3x²/2 = 0
Solve for x.
determine the dimensions of a rectangular solid (with a square base) with maximum volume if it's surface area is 289 meters
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