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'An Unusual Shape' printed from https://nrich.maths.org/
There were lots of solutions submitted and
most of you got the right answer, that the goat was tied up on a
long side of the shed, 5 feet from the bottom left corner of the
diagram in the question. There were a couple of different ways of
working this out:
Stuart has fixed the rope 5 feet from the bottom left hand
corner - This is because there is 20 feet to the top right corner
which is where the goat got to and how long the rope is.
(thanks, Thomas from Wilson's
School)
Stuart fixed the rope directly above the lowest point of
the lower semi-circle as it was the only place that would let the
goat eat all of the grass that it did on the
diagram.
(Hussein, also from Wilson's
School)
The goat is tethered 5 metres in on the side where the
majority of grass has been eaten. Therefore the goat can eat the
full half on its walls side (South side). The rope can then bend
round east for the 10 metres along the wall then north and another
10 metres along the side. On the west side the rope goes 5 metres
west then 10 metres along the west wall and then the remaining 5
metres of the rope is used eating as far as it can along the north
wall. The goat then eats the grass that it can reach in those
regions.
(a nice explanation from James, also from
Wilson's School)
Danielle from Darrington C of E Primary
School used a picture to help explain her answer.
Well done to Tim, Michael, Muntej,
Jamie, Charlie and Kartik, who all got this
right.
Kartik and Tim went on to work out the area
of grass eaten by the goat. Both of them did this by splitting the
area up into several shapes, all of which were halves or quarters
of circles, and adding the areas together.
I think the area available to the goat was 904 foot
squared. This is because if you look at the shape, you can split it
into 3 quarter circles and one semi-circle. You then work out the
area of the different parts and add them all up.
A few of you considered the problem of
whether the goat would have a larger area to graze if the location
of where it was tied changed.
Kartik provided a beautifully worked
set of examples to
show that it did vary and went on to suggest that the grazing area
is greatest when the goat is tethered at a corner.
Elliott and Oliver also showed that
tethering the goat to a corner increased its grazing
area.
If you put the goat on the bottom right corner of the shed then the
area =
$(20\times 20 \times \pi) \times 0.75 =
300 \pi$
$+ {{10 \times 10 \times \pi }\over {4}} =
{25 \pi}$
$+ {{5 \times 5 \times \pi }\over {4}} =
{6.25 \pi}$
$300 + 25 + 6.25 = 331.25 \pi$
$= 1040.65 ft^2$
Oliver from Colchester Royal Grammar School
considered a different approach to this.
The area eaten is 287.5$\pi$
$ft^2$. If there was no shed in the way the total possible area for
the goat to eat is:
$A= \pi \times 20 \times 20=400\pi$
So the shed is stopping the goat
eat an area of 112.5$\pi$ feet squared of grass. This uneatable bit
could be reduced by tying the goat at the corner of the shed
because the goat is further away from the centre of the shed.
The area of available grass is now:
$A= (3 \times {{\pi x 20 x 20}\over{4}}) + ({{\pi x 10 x
10}\over{4}}) +({{\pi x 5 x 5}\over{4}})$
$A= 300\pi + 25\pi + 6.25\pi$
$A = 331.25\pi$
which is roughly 1040.65 feet squared.
Alastair, Ed and Llewellyn from St
Peter's College also came to the same conclusion.
Aswaath's (Garden International School,
Malaysia)
solutions include a formula to
show that you get the greatest grazing area when the goat
is tethered at the corner.
But can we be sure that tethering the goat
to the corner provides it with the maximum possible grazing area
for any length of rope?
Maybe you could try to prove
this.