If 2200 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box. Volume = cubic centimeters.

Answer 1

The volume of a box is given by:

V = l•w•h

It's a square base so, let's call he side of the square "x". That means both "l" and "w" are "x" so the volume equation is: V = x²•h

The volume is to be maximized, so take the derivative and set it equal to zero to find the local max. But the volume function has 2 variables in it (both "x" and "h". Use more information to get a single variable.

It's 1800 sq cm of material. That's the surface area of the box. The surface area can be calculated by:

SA = Area_bottom + 4Area_sides

SA = x² + 4x•h

2200 = x² + 4xh

2200 - x² = 4xh

h = (2200 - x²)/4x

h = 2200/4x - x²/4x

h = 550/x - x/4

Plug "550/x - x/4" in for "h" in the volume equation:

V = x²•h

V = x²(550/x - x/4)

V = 550x - (1/4)x³

Take the derivative and set it equal to zero to maximize the volume:

V' = 550 - (3/4)x²

550 - (3/4)x² = 0

550 = (3/4)x²

550(4/3) = x²

x = about ± 27.08

The dimension can't be negative.x = sqrt(550*4/3) = about 27.08

Using the volume equation:

V = 550x - (1/4)x³ plug in x = sqrt(550*4/3) to get the volume at that maximum:

V = 550(sqrt(550*4/3) )- (1/4)(sqrt(550*4/3))³

V = about 9929.38 cubic cm.

Plug in values 28 and 27 to convince yourself it's a maximum.

V(x) = 550x - (1/4)x³

V(27) = 550(27) - (1/4)(27)³ = about 9929.25 cm³

Now x = 28

V(28) = 550(28) - (1/4)(28)³ = 9912 cm³