Bottles, glasses and shapes

D. Leela and class-mates, Chennai

"It's so boring without Nivedita," grumbled Leela, staring
disconsolately at the wall. 
"And Kavita," agreed Maya. "There's nothing to do - I've even finished
watering my plant."
"You'll kill your plant if you keep watering it all the time," warned
Leela. "Give it some time to grow!"
"Oh, no, I'm giving it exactly 200 ml every time," said Maya. 
"Where did you get a 200 ml measuring jar," asked Leela.
"I'm using a tea-cup. Amma said that was 150 ml. Then I pour some
extra."
Leela laughed. "How does that make it 'exactly' 200 ml?" she asked.
"Well, it can't be so wrong," protested Maya. "Where do I get a
measuring jar from?"
Leela sat up. "There must be some way of making one. Let's ask Amma how
she measures things."
Mother was in the kitchen. "Amma, how do you measure things exactly?"
"Length or weight?" asked Mother. 
"Actually, we want the volume of things. Like water, or maybe any
liquid."
"Why not the volume of solids as well?" asked Maya.
"Start with liquids first," said Mother. "What are the liquids that you
know come in well-known volumes?"
"Well, oil and fruit juices come both in bottles and in
plastic/tetra-packs.  Also soft drinks," said Leela. "Shall we use the
bottles as measuring jars?"
"Yes," said Maya, excitedly. "See, this water bottle says '1 litre' on
the wrapper. So we can use this as an exact measure."
"That's fine for large measures. But how will you measure 200 ml with
it?"
"Well, I can pour out the 1 litre into 5 equal glasses, so each glass
will be 200 ml," said Maya. 
"That's a  nice idea," said Leela, "but how will you tell that they are
all equal?"
"We'll line up all the glasses and make sure the water level in each
glass also lines up when filled."
Leela and Maya carefully filled the glasses from the bottle. 
"There's already an error," said Leela. "We don't know whether the
bottle measures 1 litre when fully filled to the cap or just up to the
neck."
Maya grumbled, "Manufacturers should put a mark indicating 1 litre on
the bottle! That way, you'll also know if you are getting your money's
worth!"
Leela was pouring water from one of the five glasses into a fatter
glass. 
"See, she said, "the water only comes up about half as high in this fat
glass."
Maya took a look. "You must have spilled some water," she said. "This
glass is not twice as fat as the other one. Why should the water come up
only half as high?"
"No, I didn't spill any water," protested Leela. "If you want, try it
again with another of the five glasses of water."
Maya tried it and sure enough, the water came up to half the height of
that in the other glasses. They got the idea of measuring the size of
the glass.
"The top and bottom of a glass is like a circle, so let's measure the
diameter of the two glasses and compare them," said Leela.
They got a scale and measured the glasses.
"It's just one and a half times fatter," said Maya. "So for some reason,
if you pour water from one glass into another glass that's one and a
half times fatter, the water height is only half as much in the fatter
glass."
"So, if you fill the two glasses to the same height, that means the
fatter one holds twice as much water as the thinner one," said Leela.
"Hmm, next time Mother gives me milk in the fat mug I'll know what she's
trying to do!"
"Actually, remember that tall glass of orange juice we had in the
restaurant last Saturday? I was wondering how I managed to drink that
much juice. But its height is misleading - the glass was tall and thin
so probably there was not all that much juice in it!" said Maya.
Leela was rummaging around for different glasses. Soon they had a table
of measurements.

Excess Diameter
Decrease in Height (approx)
1.25
1.5 times
1.5
2 times
2
4 times

Maya sat staring at the numbers.
"What are you thinking of, Maya," asked Leela. 
"The ocean," said Maya.
"What about it?"
"The ocean is like a big fat glass, right? And not only is it fat, it's
also high! So I was trying to imagine how many glasses it would take to
fill it."
Leela laughed. "Aren't you glad you don't have to do it? But wait, I've
been thinking about solids and I think I know how to find their volumes.
We'll simply put it in the glass of water. The water will rise and we'll
know what is the excess volume - that will be the volume of the solid."
First they measured out 200 ml of water and divided it in equal amounts
in two identical glasses so there was 100 ml of water in each glass.
Leela asked Mother for a Re. 2 coin and put it into one glass of water.
But there was hardly any change in the level of water compared with the
other glass. Leela looked disappointed.
Maya suggested, "Let's put in lots of coins. Then the water will rise
more and we'll be able to tell the volume of one coin if we measure the
volume of many."
They got ten Re. 2 coins and put them in, but the water level still
hardly changed.
Maya took all the coins that she could collect and put them in. When she
had put in 40 coins, the water in the glass finally rose about 1 cm
higher than that of the other one.
"I can't believe it; the glass appears to be full of coins, but there is
hardly any rise in the water level," she said.
"Yes, how can we measure such a small rise? If the volume had doubled,
then we would know that the coins had a total volume of 100 ml. Since we
know how many coins we have put in, we could have found out the volume
of each coin. But now ..."
"We can still find the volume. We divided 1 litre into five glasses to
measure 200 ml. Now we must just have to keep dividing 200 ml till we
get 1 cm of water in the glass," said Maya.
"I have a better idea. We have a small measuring cup, for medicine. I
remember it has 5 and 10 ml markings. That will be more accurate."
"Wonderful," said Maya. They measured out with the medicine cup and
found that the water rose 1 cm in the glass when about 30 ml of water
had been poured into it.
"So the volume of 40 coins is 30 ml, and the volume of each coin is
about 3/4 ml," said Leela, at last, when they had finished.
Leela said, "How funny. So, if you take 5 ml of medicine and flatten it
out in the shape of coins, you'll get 7 coins!"
They showed the glass of coins to Mother. Mother laughed. "What a nice
discovery," she said. "Leela, have you learned about areas in school?"
"We live in Tiruvanmiyur area," said Maya promptly.
Leela giggled. "Silly, not that area; this is maths."
"It's all the same thing," said Mother. "Tiruvanmiyur is called an area
precisely because it is an area! Only, you may have learned to find the
areas of circles and squares in class: Tiruvanmiyur is a funny shape
whose area may not be easy to find."
"Yes, I know that a circle of radius R has an area A=pR2," said Leela.
"Also, I know that p=3.14 though I am not quite sure why."
"That's good enough," said Mother. "You can understand that the volume
of water in a glass which has a circle shape of area A at the top or
bottom and a height h is the product Ah."
Leela thought for a while. That means, the volume of water depends on
the square of the radius?"
"Yes," said Mother. So, if the radius (or diameter) doubles, what
happens to the volume?"
"Well, R becomes 2R so R2 becomes (2R)2 or 4R2. Oh, the area becomes 4
times, so the volume also does. Maya, where's that table we made?"
Maya had already gone to look at it. "Yes", she said, "when the glass
was twice as fat, the height in the glass was 1/4 that of the comparison
glass."
"So, if you had the same volume, then the height will be reduced to a
quarter. Wow!"
Maya said, "Amma, if you had told us that formula, I could have told you
the volume of the coin at once. See, the coin is also like a fat glass:
it has a circle shape at the bottom and you can think of its thickness
like a height."
"That's right," said Mother. "But then you wouldn't have had the fun of
discovering this for yourselves. Also, you wouldn't have known how to
spend the Saturday afternoon!"
Leela had not been listening. "You could not have found the volume of
the coin this way, Maya. Look, it's hard to find out the thickness of
the coin."
"We learned how to do this last time," said Maya. Measure the thickness
of ten coins and then divide by 10 to get the thickness of 1 coin."
"Of course."
They found the coin had a diameter of D=2.5 cm and a thickness of h=0.15
cm. The volume of the coin was p(D/2)2 h= 0.74 ml.
"Fantastic," said the children. "Look, Amma, we got it correct. We got
3/4 which is 0.75 instead of 0.74, but that's still good, right?"
"Yes indeed," said Mother. "Come and drink your milk. As a treat, I'll
let you choose whether you want a tall, thin glass, or a short fat one!"
Notes: Glasses may not be uniform. In that case, take the average area
of the bottom and top of the glass. Also, remember that 1 ml = 1 cc, so
if you take all measurements in cm, you'll get the volume in ml; 1000 ml
= 1 litre.