find the largest possible volume of the box

Question

If 1800 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=?

Answer 1

Have you drawn a figure?

 

A box with a square base and open top...  x = sides of square.  y = vertical height

 

The base has dimensions (x)(x) or its area is x².

Each side has an area of (x)(y) and there are four of these.  So the sides all together have an area of 4xy.

 

The total area of material needed is x² + 4xy and it has to = 1800 square centimeters.  This is your secondary equation. 

 

The primary equation is the equation that you want to maximize or minimize.  In this case you want to maximize volume, so you will need the volume of the box.

 

V = x²y

 

How will you get this into a single variable?   Use the secondary equation  x² + 4xy = 1800

 

It would be easiest to solve the secondary equation for y, then plug it into the equation for volume.

 

V = x² [(1800-x²)/4x]   simplifies to

 

V = [(1800x² - x⁴)/4x]     factor an x out of the top and out of the bottom

 

V = [(1800x-x³)/4]

 

 

NOW you can take the deriv, set it = 0, find the x that maximizes the volume, then find the volume since the problem is asking for volume, not a length of the side of the base.