Given that the product of the area and the perimeter of a rectangle is at least 1, what is the smallest possible value of the perimeter?
For a rectangle of a given area A, the smallest perimeter P is given
by a square*
Since we wish to minimize P under the condition A P >=1, we use a square of side s. Since both A and P grow with s, we choose
A P = 1.
A P = s^2 (4 s) = 1
s^3 = 1/4
s = 2^(-2/3) ~ 2.996052
P = 4 s = 2^2 * 2^ (-2/3) = 2^(-4/3)
P ~ 2.5198
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* For a rectangle of area C^2, prove a minimum perimeter results
from a square of sides s1 = s2 = C.
C^2 = s1 s2. Call s1 = x, C^2 = (s1) (s2) = (x) (C^2 /x),
P= x + C^2/x,
Find minimum:
dP/dx = 1 - C^2 x^(-2) = 0, when x = C
QED
Edited on October 10, 2024, 3:23 am
soln