let-x-and-y-be-positif-real-number-such-that-1-x-y-9-and-x-2y-3x-what-is-the-largest-value-of-9-y-9-x-

Question Number 80347 by jagoll last updated on 02/Feb/20

l e t x a n d y b e p o s i t i f r e a l n u m b e r

s u c h t h a t 1 ⩽ x + y ⩽ 9 a n d

x ⩽ 2 y ⩽ 3 x . w h a t i s t h e

l a r g e s t v a l u e o f \frac{9 - y}{9 - x}

Commented by john santu last updated on 02/Feb/20

i t h i n k y o u c a n s o l v e b y

l i n e a r p r o g r a m m i n g

Commented by mr W last updated on 02/Feb/20

w h a t i f t h e q u e s t i o n i s

w h a t i s t h e l a r g e s t a n d s m a l l e s t v a l u e

o f x^{2} + x y - 2 x - 3 y ?

Commented by jagoll last updated on 03/Feb/20

b y n o n l i n e a r p r o g r a m m i n g

h a h a h a h a

Commented by mr W last updated on 03/Feb/20

l e t k = x^{2} + x y - 2 x - 3 y

\Rightarrow x^{2} + x y - 2 x - 3 y - k = 0

t h i s i s t h e e q u a t i o n o f a h y p e r b o l a .

m a k i n g t h i s c u r v e t o u c h t h e a r e a

g i v e n, w e c a n f i n d t h e m i n i m u m

a n d m a x i m u m v a l u e o f k .

k_{m i n} = - \frac{169}{40}

k_{m a x} = 33

Commented by jagoll last updated on 04/Feb/20

s i r k_{m i n} a t p o i n t ?

Commented by mr W last updated on 04/Feb/20

k_{m i n} i s w h e n t h e c u r v e j u s t t o u c h e s t h e

t o p m o s t l i n e 4 .

Answered by mr W last updated on 02/Feb/20

1 ⩽ x + y ⩽ 9

\frac{x}{2} ⩽ y ⩽ \frac{3 x}{2}

t h e a r e a o f x, y i s d e f i n e d b y 4 l i n e s :

l i n e 1 : x + y = 1

l i n e 2 : x + y = 9

l i n e 3 : y = \frac{x}{2}

l i n e 4 : y = \frac{3 x}{2}

l e t \frac{9 - y}{9 - x} = \frac{y - 9}{x - 9} = k

\Rightarrow y = 9 + k (x - 9)

k i s t h e i n c l i n a t i o n o f a l i n e p a s s i n g

t h e p o i n t S (9, 9) a n d t h e a r e a d e s c r i b e d

a b o v e .

w e s e e k i s m a x i m u m w h e n t h i s l i n e (A)

p a s s e s t h e p o i n t P a n d k i s m i n i m u m

w h e n t h i s l i n e (B) p a s s e s t h e p o i n t Q .

p o i n t P = i n t e r s e c t i o n o f l i n e 3 a n d 2 :

y = \frac{x}{2}

x + y = 9

\Rightarrow x = 6, y = 3 \Rightarrow P (6, 3)

k_{m a x} = \frac{9 - 3}{9 - 6} = 2

p o i n t Q = i n t e r s e c t i o n o f l i n e 4 a n d 2 :

y = \frac{3 x}{2}

x + y = 9

\Rightarrow x = \frac{18}{5}, y = \frac{27}{5} \Rightarrow Q (\frac{18}{5}, \frac{27}{5})

k_{m i n} = \frac{9 - \frac{27}{5}}{9 - \frac{18}{5}} = \frac{2}{3}

\Rightarrow \frac{2}{3} ⩽ \frac{9 - y}{9 - x} ⩽ 2

Commented by mr W last updated on 02/Feb/20

Commented by jagoll last updated on 02/Feb/20

t h a n k y o u m i s t e r

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