Draw-a-rectangle-of-maximum-perimeter-by-ruler-and-compass-when-area-is-ab-AB-a-CD-b-are-given-

Question Number 3830 by Rasheed Soomro last updated on 21/Dec/15

D r a w a r e c t a n g l e o f m a x i m u m p e r i m e t e r,

b y r u l e r a n d c o m p a s s, w h e n a r e a i s ab .

(AB = a, CD = b are given .)

Commented by prakash jain last updated on 22/Dec/15

Area a b, let us say one side is x

Perimeter = 2 (\frac{a b}{x} + x)

\lim_{x \to 0} (\frac{a b}{x} + x) \to \infty

There is no maximum perimeter for given

area .

Commented by Yozzii last updated on 22/Dec/15

P (x) = 2 (\frac{c}{x} + x) \Rightarrow \frac{d P}{d x} = 2 (1 - \frac{c}{x^{2}}) = 2 (1 - {c x}^{- 2})

\frac{d P}{d x} = 0 \Rightarrow x = \pm \sqrt{c} ∵ x > 0 \Rightarrow x = \sqrt{c}

\frac{d^{2} P}{{d x}^{2}} = 2 (2 {c x}^{- 3}) > 0 i f x > 0

A t x = \sqrt{c} = \sqrt{a b}, \frac{d^{2} P}{{d x}^{2}} > 0 \Rightarrow m i n i m u m p t .

f o r P e x i s t s o n l y s i n c e x > 0 .

Commented by Rasheed Soomro last updated on 22/Dec/15

T han k^{S}!