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.

Answer 1

The volume of a box is given by:

V = l•w•h

In this case, they tell us we have a square base so, let's let the side of the square be "x". That means both "l" and "w" are "x" so our volume equation is: V = x²•h

Since we are trying to maximize the volume, we are being asked to take the derivative and set it equal to zero to find the local max. But our volume function has 2 variables in it (both "x" and "h" and we haven't learned how to deal with that as yet. So let's use more information given to us to see if we can get this equation down to a single variable.

We are told that we have 1800 sq cm of material. That would equate to the surface area of the box. The surface area can be calculated by:

SA = Area_bottom + 4Area_sides

SA = x² + 4x•h

1800 = x² + 4xh

1800 - x² = 4xh

h = (1800 - x²)/4x

h = 1800/4x - x²/4x

h = 450/x - x/4

So we can plug "450/x - x/4" in for "h" in our volume equation:

V = x²•h

V = x²(450/x - x/4)

V = 450x - (1/4)x³

Now, we can take the derivative and set it equal to zero to maximize the volume:

V' = 450 - (3/4)x²

450 - (3/4)x² = 0

450 = (3/4)x²

450(4/3) = x²

x² = 600

x = ± 24.495

We can throw out the negative answer since we can't have a negative dimension on the box. so x = 24.495

Using our volume equation:

V = 450x - (1/4)x³ we can plug in x = 24.495 to get the volume at that maximum:

V = 450(24.495) - (1/4)(24.495)³

V = 7348.5 cubic cm.

Just to make ourselves feel OK about this, let's double check a couple of things. Let's figure out the volume if x = 20 to see where we are:

V(x) = 450x - (1/4)x³

V(20) = 450(20) - (1/4)(20)³ = 7000 cm³

Now, lets see what it is when x = 30

V(30) = 450(30) - (1/4)(30)³ = 6750 cm³

So our answer looks good.