Expert: Paul Klarreich - 7/1/2007

Question
find dimensions of a rectangle with area 1000m2 whose perimeter is as small as possible

Answer
Questioner:   candace
Category:  Calculus
 
Subject:  calculus optimization
Question:  find dimensions of a rectangle with area 1000m2 whose perimeter is as small as possible
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Hi, Candace,

General scheme for these optimization problems:

1. Determine the variables in the problem -- the things you can control.  Try to find an "interval of meaningfulness" for them.
2. Write the quantity to be optimized in terms of the variables.
3. If necessary, eliminate all but one variable.
4. Find the critical points for the quantity -- stationary points and endpoints.
5. Pick your max or min.

In this example, you can control the dimensions, so give them names:

h = height of the rectangle.
w = width .................

Certainly h and w must be positive, but if the area is to be 1000, either could be arbitrarily large, so  h,w in [0,infinity]

You want to minimize the perimeter:

P = 2h + 2w.

Use the fact that area = 1000 to eliminate a variable:

hw = 1000
w = 1000/h = 1000h^-1

P = 2h + 2000h^-1

Differentiate:

dP/dh = 2 - 2000h^-2 = 2 - 2000/h^2

Set that = 0:

2 - 2000/h^2 = 0
2h^2 = 2000
h^2 = 1000
h = sqrt(1000)  << first dimension
w = 1000/sqrt(1000) = sqrt(1000)  << other dimension

i.e. the rectangle is a square.