Math CentralQuandaries & Queries


Question from jasmin, a student:

A farmer staked his goat at the corner of the barn 30 m long and 25 m wide so that the goat could eat the grass near the corner of the barn. He found that if the rope by which the goat was tied was lengthened by 10 m, the goat could graze over four times as much as the area. How long was the original rope, if in neither case it was as much as 25 m long?

Hi Jasmin.

If you draw the diagram for this, you will see that the goat in both cases is grazing three quarters of a circle.

Are you sure the original question says "over" four times as much? If so, then there won't be just one answer to your question. For example, a goat with a 1 cm original rope and a goat with a 10m + 1 cm new rope have much more than four times as much grass to munch. But so does a 2cm original...

What is the formula for the area of a circle? it is π r2.

So let r be the length of the original rope and a be the area she could graze with it. Let A be the area she could graze with the longer rope (it is r + 10 units in length).

This means you have two equations:

(3/4) π r2 = a
(3/4) π (r+10)2 = A

In fact, you can just divide the top equation by the bottom equation:

((3/4) π r2 ) / ( (3/4) π (r+10)2 ) = a / A

But since we know that A is at least 4 times a, then a / A < a / (4a) = 1/4.

So simplifying the equation be substituting this in and canceling terms on the left side, we have this:

r2 / ( r + 10)2 < 1/4

You should be able to solve this for r. If you don't like the < sign in there, then make it an equal sign, solve for r, then reason out from the context of the question if the rope could be longer or shorter than the value you got for r.

Stephen La Rocque.

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