Question

if a field has perimeter p, what shape maximizes the area? How can I justify my answer?

Answer 1

you could compare the ratio of Area to Perimeter for shapes. It should be fairly easy to show that the shape must be convex.

Example: for an equilateral triangle

\[\frac{ A }{ P }=\frac{ \frac{ 1 }{ 2 } b\cdot \frac{ b \sqrt{3} }{ 2 } }{3b }=\frac{ b \sqrt{3} }{ 12 }=\frac{b}{4\sqrt{3}}\]

for a square:

\[\frac{ A }{P }=\frac{ b^2 }{4b } =\frac{ b }{ 4 }\]

Answer 2

it ends up being a circle but to show that you really need calculus. is this a calculus class?

Answer 3

This is a problem solving math class. This is a really hard problem to figure out.

Answer 4

have you taken calculus?

Answer 5

I took calculus in High school and that was 5 years ago. I decided to go back to school and I have to take this class due to my major.

Answer 6

so a regular polygon will maximize the area for a given perimeter, for a given polygonal shape.

for example, a quadrilateral with the largest area, given a fixed perimeter, is a square.

Area of a regular polygon = \(\frac{1}{2}\cdot \text{apothem} \cdot \text{perimeter}\)

Answer 7

to maximize the area you must maximize the apothem. the apothem is the radius of the inscribed circle. As the number of sides increases, the shape becomes a circle and the apothem becomes the same as the radius of the circumscibed circle.

Answer 8

does that help?

Answer 9

you there?

Answer 10

yes
sorry

Answer 11

it helps a little. Thanks!

Answer 12

hopefully professor can go over it wednesday.

Answer 13

what part(s) are troubling you?

Answer 14

first of all the word apothem