Find the dimensions of a rectangle with area 1,000 m2 whose perimeter is as small as possible. (if both values are the same number, enter it into both blanks.)

The area is:

A = x * y = 1000

The perimeter is:

P = 2x + 2y

The perimeter as a function of x is:

P (x) = 2x + 2 (1000 / x)

Rewriting:

P (x) = 2x + 2000 / x

Deriving:

P ' (x) = 2-2000 / x ^ 2

We match zero:

0 = 2-2000 / x ^ 2

We clear x:

2000 / x ^ 2 = 2

x ^ 2 = 2000/2 = 1000

x = root (1000)

x = 10raiz (10)

We derive for the second time:

P '' (x) = 4000 / x ^ 3

We evaluate x = 10raiz (10)

P '' (10raiz (10)) = 4000 / (10 * root (10)) ^ 3 = 0.126491106> 0 (it is a minimum)

The dimensions are:

x = 10raiz (10)

y = 1000 / (10raiz (10)) = 100 / (root (10)) = 100raiz (10) / (root (10) * root (10))

y = 100raiz (10) / (10)

y = 10raiz (10)

Answer:

the dimensions of a rectangle with area 1,000 m2 whose perimeter is as small as possible are:

x = 10raiz (10)

y = 10raiz (10)