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.)

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Jaderoyce1978 Jaderoyce1978

14-04-2018
Mathematics

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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.)

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carlosego carlosego · Answer 1 · 2018-04-23T17:34:32+02:00

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)