The Stingy Rancher And His Fence

Calculus Level 4

A rancher has been given a square mile of land, a perfect square with a perimeter of 4 miles. He decides he is going to fence in his property, but fencing costs money, and so he's only interested in fencing in the most land he can for the cost of the fencing, which is $1000 per mile. Let $A$ be the area of the land fenced in, and $B$ denote the total cost of the fence. When $A/B$ is maximized, how much did he pay for fencing, expressed as $\left\lfloor B \right\rfloor ?$

Details and Assumptions

There is no existing fencing on the land given to the rancher, including at the boundary.
Because the area is bounded, $\frac{A}{B}$ will be bounded above. For example, we are not allowed to use a circle of radius 1.

This problem was suggested by Open Question...

The answer is 3544.

1 solution

Michael Mendrin
May 27, 2014

Let $x$ be the radius of the quarter circles. Then the fenced in area divided by the perimeter is:

$\dfrac { \pi { x }^{ 2 }+4x(1-2x)+{ (1-2x) }^{ 2 } }{ 2\pi x+4(1-2x) }$

Using calculus, the maximum is found when $x=\dfrac { 2-\sqrt { \pi } }{ 4-\pi }$ , in which the total perimeter works out to $2\sqrt { \pi } =3.54491..$ , so that the farmer paid $\$3544.91$ for his fence. The answer then is $3544.$

Well, I don't know how you deduced the shape. I think you should have mentioned what was the shape going to be!

Avineil Jain - 7 years ago

By the isoperimetric theorem, between 2 fixed points, it's an arc of a circle that for a given arc length it bounds the maximum area. Secondly, any such perimeter of minimum length and maximum bounded area will have no discontinuities. This leads to quarter circles at the corners of this problem. Symmetry takes care of the rest.

Michael Mendrin - 7 years ago

All right! I was assuming a whole circle that would have been inscribed in the square. Therefore, my answer was coming $3542 !

Avineil Jain - 7 years ago

@Avineil Jain – If the rancher had fenced in a perfect circle of diameter 1, the cost of the fence would have come to $3141.59. Probably a lot of people jump to this conclusion.

Michael Mendrin - 7 years ago

Well, Michael Mendrin I must say you did it very well

Muhammad Shafaat Nawaz - 7 years ago

That's interesting!

Calvin Lin Staff - 7 years ago

I didn't even get that. Did you? :P

Finn Hulse - 7 years ago

The problem, not the solution. The solution was fine.

Finn Hulse - 7 years ago

Nice solution. But how to deduce/prove that it is this shape that maximizes the area/expense ratio? Once you identify the shape, finding its area and perimeter and maximising their ratio is relatively easier.

Surya Narayanan - 7 years ago

See above.

Michael Mendrin - 7 years ago

My Excel Spread sheet program yields $3563.66. with a sector angle of 26.46 degrees. ???

Guiseppi Butel - 6 years, 10 months ago

The Stingy Rancher And His Fence

The answer is 3544.

1 solution

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