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Find Depth of Water in a Tank

Date: 08/02/2003 at 04:16:59
From: Hasan
Subject: Volume

A rectangular tank measures 4m long, 2m wide and 4.8m high. Initially 
it is half full of water. Find the depth of water in the tank after 
4000 litres of water have been added to it.

Date: 08/02/2003 at 22:18:36
From: Doctor Ian
Subject: Re: Volume

Hi Hasan,

Try thinking of it this way. Imagine there are two tanks instead of
one. They're the same size, but one is full, and the other is empty. 

What size are they? They're each half as tall as the single tank in
your problem, so the dimensions are

  length = 4 m
  width  = 2 m
  height = 2.4 m

Here's a picture of the situation:

         Old problem
  2m  /               /|
     /               / |
    +---------------+  | 4.8 m
    |               |  |
    |               |  |
    |               |  +
    |               | /
    |               |/

         New problem
  2m  /               /| 2.4m
     /               / |
    +---------------+  + 
    |               | /| 2.4m
    |               |/ |
    +---------------+  +
    |               | /
    |               |/

So now we want to know: If we put 4000 liters of water into the top
tank, how high will the water level be? And when we figure that out,
we can add 2.4 meters to it to get the answer to _your_ problem. 

Does this make sense? 

Okay, so how do we get the height of the water in the top tank?  Well,
we can change the problem a little bit again. What if we have a tank
that holds exactly 4000 liters of water? If the length is 4 meters, 
and the width is 2 meters, how high will that tank be?  

The volume of a rectangular prism (which is the name of this kind of
shape) is 

  volume = length * width * height

We know the volume: It's 4000 liters. And we know the length and 
width. So 

  4000 liters = 4 meters * 2 meters * ? meters

But there's a problem!  On the right, we get a volume of 

  (4 * 2 * ?) cubic meters

while on the left, we have the same volume expressed in liters. What
you'll want to do is find out how many liters are in a cubic meter,
and convert:

                1 cubic meter
  4000 liters * ------------- = 4 meters * 2 meters * ? meters
                ?? liters

And then you just have to figure out what value of '?' makes the
equation true.  When you have that, you know the height of a tank that 
would hold exactly 4,000 liters.  That's the same as the height of the 
water level in the top tank, and if you add 2.4 meters to that, you 
get the water level in the original tank.  

Were you able to follow all this? It illustrates an important point,
which is that most of the time, when you're solving a math problem,
what you're looking for isn't an immediate answer, or a formula that
can get you the answer in one step. What you're looking for is a way
to change the problem you have into one that would be simpler to 

In this case, we did that twice.  First, we didn't want to worry about
the water that was already there, so we pretended it was in a separate
tank.  Then, we didn't want to worry about water level separately from
the volume of the tank, so we tried to find the volume of a tank that
could hold _only_ 4000 liters of water.  

Does this help?  Write back if you'd like to talk more about this, or
anything else. 

- Doctor Ian, The Math Forum 
Associated Topics:
High School Polyhedra
Middle School Polyhedra
Middle School Terms/Units of Measurement
Middle School Word Problems

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