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You may have heard the term 'Calculus', it's a big idea in
mathematics and so powerful that I find it hard to imagine
mathematics without it. It's main technique, differentiation, makes
problems like this fairly easy  with a little care over the
algebra. So if you are thinking about continuing with mathematics
beyond Stage 4, learning about differentiation would be one of the
big benefits that await you.
But differentiation is not really Stage 4 mathematics, though
very close, so we don't use it in solutions because that would be
unfair to students who haven't seen the idea before, it would be
like suddenly changing into a language you haven't had a chance to
learn yet.
If that's got you interested why not take a look at Vicky Neale's
article :
Introduction
to Differentiation
And thankyou to David and Berny from
Gordonstoun and to Naren from Loughborough Grammar School who sent
in great solutions using that technique .
In fact the use of differentiation did more than get an answer. It
found the result that the cone needed to have its height 1.41 times
bigger than its radius, like the Stage 4 result below, but found it
in this more interesting form.
And the value of that is that it suggests a new direction to
pursue with this problem : it seems so neat.
Why the square root of 2 ? Is there a connection with the
diagonal of a square ? Or is it something else ?
That's maybe a little bit on from us at
Stage 4 so here's a way to solve a problem like this using Stage 4
mathematics. It's really 'trial and improvement' but using a
spreadsheet to make the calculation effortless. .
We are going to use :
 the radius, which we'll adjust to get nearer and nearer to the
answer,

the height, which will be determined by our choice of radius, so
that we get the chosen target volume

the slant length, which we'll find using r and h and
Pythagoras,

and the surface area, for which there's a great little
formula.
The volume of a cone is one third the volume of a cylinder with the
same base and height.
If we took the target volume to be 1 litre, 1000 ml, then the
formula that connects h and r is
The slant length (s), using Pythagoras, is
And the surface area of a cone is
plus the base if you need it  here we don't.
Incidentally, if you don't know where that surface area formula
comes from it may be good to take a moment to look at that. Flatten
the curved surface out to get a sector (how do you know it's a
sector ?). The radius will be the cone's slant length, so you can
calculate the area of the whole circle. To know the proportion that
the sector is of that circle compare the sector arc, which is the
cone's base circumference, with the circumference of this new
circle, radius s.
Back to the funnel and using as little plastic as possible.
Take a look at this spreadsheet :
Funnel
Can you see what each column does ? Click on a cell and check
the formula.
 The first column has increasing radius values which you can
control.
 The next column calculates the height, because we knew the
volume was 1000 ml
 The radius and height are then used to calculate the slant
length.
 And the final column uses the radius and the slant length to
calculate the surface area, which we want to be as small as
possible
There's even a graph so you can have some sense for how surface
area varies as the radius value ranges across your chosen interval.