A rectangular 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=yx ²

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