Let f(x) be the perimeter of a rectangle with an area of 9 units^2 and one side with length x.

Question

Let f(x) be the perimeter of a rectangle with an area of 9 units² and one side with length x.

f(x) = ?

What is the minimum perimeter of all rectangles with this area?

Perimeter =

Answer 1

sqr9 = 3

3x4 = 12 inches = minimum perimeter for a rectangle with area = 9 in^2

minimum perimeter of a rectangle for a fixed area is with a square. for A=9, the square's side = sqr9=3

Perimeter = 4 times the side of a square = 4x3 = 12 in.

or use differential calculus

A=Lw = Length times Width for a rectangle

P= 2(L+w) = Perimeter of a rectangle

L=A/w = 9/w

P = 2(L+w) = 2(9/w+w)

take the derivative and set =0

P' = 2[-9/w^2 +1) = -18/w^2 +2 = 0

-18 +2w^2 = 0

2w^2 = 18

w^2 = 9

w= + or - 3

L= 9/+or-3 = + or -3

dimensions of the rectangle with minimum perimeter are

either 3 and 3 inches or -3 and -3 inches.

Minimum perimeter = 4(3) = 12 inches

or 4(-3)= -12 inches

-12 inches is actually a smaller perimeter, but generally you assume dimensions are positive, ignoring the negative solutions. But in an Einsteinian world with negative time, parallel universes, or the twilight zone, maybe -3 by -3 inches has the minimum perimeter.

It's similar to a negative number joke

3 scientists were sipping coffee at an outdoor cafe with a view of a clearly empty building across the street. They all watched as 4 people entered, then later 2 people exited. The scientists gave their expert opinion explaining this unexpected incident. The physicist said the 2 people who disappeared must have gone through a wormhole in the space time continuum, coming out in an alternative universe. the biologist said no, what happened was an unusually speedy aging and decay, leaving only 2 survivors. The mathematician said obviously not. What happened was the empty building was very empty, originally containing negative two people. then when 4 entered, there were -2+4 = 2 people left. Just simple 3rd grade arithmetic solves the problem. The physicist and biologist rolled their eyes and mockingly said, Is that what you teach your 3rd grade students? No wonder math scores are way down in this country.