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Rasek [7]
3 years ago
15

A rancher wants to build a rectangular pen with an area of 36 m2.

Mathematics
1 answer:
MakcuM [25]3 years ago
3 0

a).  If the width is 'w' and the area is 36, then the length is  36/w.
The amount of fencing required is 2 lengths + 2 widths (the perimeter).
That's  <em>2w + 72/w</em>  or  <em>(2/w)(w²+36)</em>  or<em>  2(w + 36/w) </em>.

b).  The shape that requires the minimum amount of fencing is a circle
with area = 36 m² .  The radius of the circle is about 3.385 meters, and
the fence around it is about 21.269 meters.

If the pen must be a rectangle, then the rectangle with the smallest perimeter
that encloses a given area is a square.  For 36 m² of area, the sides of the
square are each <em>6 meters</em>, and the perimeter needs 24 meters of fence to
enclose it.

I don't know how to prove either of these factoids without using calculus.


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