Arectangular solid with a square base has a surface area of 337.5 square centimeters. (let x represent the length of the sides of the square base and let y represent the height.)
x=
y=
max volume=

The surface aera will be the 2 square bases plus the 4 sides or
2x²+4xy
the surface area is 337.5
2x²+4xy=337.5
divide both sides by 2
x²+2xy=168.75

now the volume
V=LWH
V=yx²
now we just need to solve for y in other equation so we can subsitute that for y

x²+2xy=168.75
minus x² both sides
2xy=168.75-x²
divide both sides by 2x
y=84.375/x-x

subsitute that for y in other equation

V=yx²
V=(84.375/x-x)x²
V=84.375x-x³
take derivitive to find max (or just graph and find where max is)
V'=84.375-3x²
it equals 0 at x=√28,125 and at x=-√28,125 
the sign changes from positive to negative at x=√28,125
so x=√28,125≈5.3033008588991
find y

sub back

y=84.375/x-x
using math
y≈10.606601717798

max volume=yx²=298.31067331307 cubic centimeters