Welcome to NRICH.

 
Tethered horses and barns
By The Editor:

This is a combination of two separate threads about the same GCSE coursework. If you are also doing this problem as coursework, please remember that you need to declare any help you have had. This will not necessarily affect your mark, particularly as we try to give only hints.

By Daisy Acres (P2264) on Monday, March 27, 2000 - 09:51 pm:

If you had a barn 20m by 10m and a horse was tethered with a rope to the centre of the long side, when the rope was longer than 30m how is the area the horse can reach calculated? I'm stuck as I can't figure out how to calculate the area which overlaps (you'll see what I mean if you draw a diagram). Please help!

By Sean Hartnoll (Sah40) on Wednesday, March 29, 2000 - 07:51 pm:

The region the horse can reach in theory is given by a semicircle radius 30m. Some of this will be outside the barn. I suggest drawing the semircle on top of a drawing of the barn. You will get a load of triangles and segments of a circle. You can then calculate all the areas the horse can reach by adding up the areas inside the semicircle which are also inside the barn. You will need the formula for the area of a circle segment, the area of a triangle (note that all the triangles you get will be right-angled triangles or isosceles triangles), and you will also need to use Pythagoras's theorem to find the sides of the triangles. Hope you can work it out from here!

Sean

By The Editor:

I think Sean is assuming that the horse is inside the barn, when it may in fact be tethered to the wall outside.

Probably the best way to get a feel for what is happening is to get some string.

Here is a diagram showing what happens with a 15m rope. The bottom part of the region is part of a circle whose radius is the length of the rope. But to the right of the barn, the radius is different: can you see what it will be?


For your problem, things get a little more complicated, because you're going to be able to get round the top of the barn as well as the side. Fiddle with some string, and I expect you'll be able to work out what to do.

By Sim Chana (P2137) on Sunday, March 5, 2000 - 06:03 pm:

here's the first part of the question... "Jack keeps his horse in a large field with a big barn in the middle. This barn is 20m long and 10m wide. The horse is tied to a point on the middle of one of the longer sides of the barn. How long should the rope be so that the horse can graze 150m2 of grass?"

I managed to figure this one out and I think the answer is 9.77m (3.s.f)... however, since this is an investigation the next part of the question says, "Investigate the area of the grass the horse can graze for conditions that you set."

To get into the highest strand for my higher GCSE I need @ least 3 variables but I am having problems thinking of plausable ones... I have thought of the following:
*changing the dimensions of the barn
*tethering the horse @ the top of the barn
*changing the area to be grazed

I can't think of more and I need more to choose from so can you please help me out there??

By Harry Smith (Harry) on Monday, March 6, 2000 - 09:31 am:

Dear Sim

First of all, well done. 9.77 metres is the right answer to the first part of your question.

While investigating this problem it is probably best to change the variables, then work out how much area the horse can graze. Once you have an idea of how this works, you can ask yourself questions that begin with an area and then ask you for the right length of rope.

The first thing to think about changing is the length of the rope. The problem becomes much more interesting once the rope is more than 10 metres long (can you see why?). Try ropes 15, 20 and 25 metres long and work out the area the horse can graze. Try answering questions like... "How long should the rope be so that the horse can graze 200m2 of grass?". Think about what happens if the rope is more than 30 metres long. Try and work out a formula which will tell you the area which can be grazed from a
rope which is x metres long.

Now you've had a good look at the problem in its given form, try changing a few of the variables you mention... where the horse is tethered, how big the barn is. Once you've changed one of these variables, go back and look at the problem again, seeing what area can be grazed with different lengths of rope.

What you've mentioned already is probably enough for you to achieve the highest strand provided you are thorough with your investigation. If you're feeling brave though, here are some other possibilities:

change the shape of the barn - What if it is a triangle or a hexagon? What if it is some irregular shape of your choosing?

introduce fences - There could be a straight fence 5 metres long coming out of one corner of the barn. How does the fence affect the area which can be grazed? What if the fence was curved, or had a corner or hole in it?

This one is a real extension, but if you really want to impress is could be fun:

change the dimension - Imagine a sea monkey is tethered to the bottom of a swimming pool full of chocolate. What volume of chocolate can it eat (ignore the fact that when it eats the chocolate more will fill up the space!)

Good luck. Let us know how you get on.

Harry