) If 1900 square centimeters of material is available to make a box with a square base and an open top,?

Question

find the largest possible volume of the box.

Answer 1

the height of the box will be x the area of the base of the box will be (y-2x)^2
the equation to find the area of the material is (2x + y)^2 = 1900 so solve for y
y= 1900^1/2 - 2x
substitute this into the equation for the base of the box, (1900^1/2 - 4x)^2
for the formula for the volume of the box multiply the area for the base time the height, which is x
V= x (1900^1/2 - 4x)^2
take the derivative of this to find the maximum
once you find the maximum value of x you have the height now solve for y to get a side
then since the volume will be equal to the area of the base times the height multiply y^2 by x
or just sub x back into the volume equation if u don't feel like solving for y to get the volume
(i think i did the algebra right but u may want to check it to be sure)

Answer 2

3061.257408 Cubic centimeters

The box would be made from a sheet approximately 43.58 Cm square. Thus each side of the box would be approx 14.52 CM across. Thus the volume would be
14.52 * 14.52 * 14.52 = 3061.257408