Converting Volumes, Such As Cubic Inches To Cubic Feet

Date: 02/09/2004 at 13:12:51
From: Cindy
Subject: Buying correct amount of sand to fill sandbox.

First of all, I'm not a student, just a stumped adult.  I would like
to fill a sandbox with 3 inches of sand.  The sandbox measures 25ft X
80ft.  How much sand do I need to buy?

I've tried using the volume formula by converting feet to inches, and 
then checking my answer by converting inches to feet.  I get two
different answers, so I'm doing something wrong.  Here's what I have:

 25 ft = 300 inches  L
 80 ft = 960 inches  W
           3 inches  D

Total volume is 300*960*3 = 864,000 cubic inches or 72,000 cubic feet
or 24,000 cubic yards.

Now, using feet -

  25ft = 25ft
  80ft = 80ft
   3in = .25ft

Total volume is 25*80*.25 = 500 cubic feet or 166.6 cubic yards.

What am I doing wrong here?

Date: 02/09/2004 at 18:27:18
From: Doctor Jeremiah
Subject: Re: Buying correct amount of sand to fill sandbox.

Hi Cindy,

The total volume is indeed 864000 cubic inches.  It's your conversion 
back to cubic feet that doesn't work, and there's a good reason 
why.  Imagine a cube 12 inches x 12 inches x 12 inches.  That cube is 
only 1 cubic foot in size, but its volume is 12x12x12 = 1728 cubic
inches.  To convert 864000 cubic inches to cubic feet you must divide
by 12x12x12.  And 864000/1728 = 500 cubic feet.

A cube that is 3 feet x 3 feet x 3 feet is only 1 cubic yard so to 
convert 500 cubic feet to cubic yards you must divide by 3x3x3.  And 
500/27 = 18.5 cubic yards.

Here is another way to look at it:

     25ft        80ft      .25ft         (25 x 80 x .25) cubic ft
  --------- x --------- x --------- = -----------------------------
  3 ft/yard   3 ft/yard   3 ft/yard   (3 x 3 x 3) cubic ft/cubic yd

Now it's a little more obvious why you need to divide by 3x3x3 when 
you are converting cubic feet to cubic yards.  And the same thing 
could be done for cubic inches to cubic feet like this:

    300in      960in      3in        (300 x 960 x 3) cubic in
  -------- x -------- x -------- = -----------------------------
  12 in/ft   12 in/ft   12 in/ft   (12 x 12 x 12) cubic in/cubic ft

Let me know if you have more questions about this.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 02/09/2004 at 21:48:33
From: Cindy
Subject: Thank you (Buying correct amount of sand to fill sandbox.)

Thank you so very much for your wonderful response!  I appreciate your
assistance!