Euler found four whole numbers such that the sum of any two of the
numbers is a perfect square. Three of the numbers that he found are
a = 18530, b=65570, c=45986. Find the fourth number, x. You could
do this by trial and error, and a spreadsheet would be a good tool
for such work. Write down a+x = P^2, b+x = Q^2, c+x = R^2, and then
focus on Q^2-R^2=b-c which is known. Moreover you know that Q >
sqrtb and R > sqrtc . Use this to show that Q-R is less than or
equal to 41 . Use a spreadsheet to calculate values of Q+R , Q and
x for values of Q-R from 1 to 41 , and hence to find the value of x
for which a+x is a perfect square.
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
Find the frequency distribution for ordinary English, and use it to help you crack the code.
There is a Stage 5 topic called
Differentiation (part of Calculus) which makes these sorts of
problems rather easy.
At Stage 4 the challenge is to handle the
algebra well (the formulae that calculate, height or diameter and
then surface area) and to use a trial and improvement method to get
as close as we wish to the answer.
Using a spreadsheet takes a lot of the labour
out of trial and improvement, and also helpfully makes an automatic
table of results.
The volume of a cylinder is found from
the circular cross-section multiplied by the distance over which it
continues, and here the volume has to be 330 ml.
So when the can's
diameter is 6 cm (r=3) it's height (h) is 11.67 cm.
And when it's height
is 10 cm, the can's diameter has to be 6.48 cm
We notice that if
the volume is fixed then whether we know the can's diameter or it's
height, the other of these two will be easy enough to calculate
So whichever we already have, either r or h,
we can find the other and then use both in the surface area
That's two circles, top and bottom, plus a
rectangle (can circumference by can height - like peeling the label
off a tin).
We are now going to work systematically to
find the diameter and height of the can (volume 330 ml) which has
the least surface area.
We could start with a radius of, say, 1 cm and
find the height which makes a volume of 330 ml, then use both those
r and h values to calculate the surface area for that can.
We could then step up the radius to 2 cm and
repeat the calculations.
In fact we could continue to increase the
radius and make up a table of the surface area results each
Alternatively we could use the same method but
start with a value for height instead, and increase that.
Once we see how the surface area changes with
radius (or height) in a general way we can refine our choice of an
r or h value until, by trial and improvement, we get as near as we
please to the lowest value for the surface area.
Here's the table Sam made :
For those who like to see nice things
done with spreadsheets in mathematics here's a link to an Excel
file that does all that calculation in an instant : Cola Can