An-oil-can-is-to-be-made-in-form-of-a-right-circular-cylinder-that-be-inscribed-in-a-sphere-of-radius-R-obtain-the-maximum-volume-of-the-can-

Question Number 6430 by sanusihammed last updated on 27/Jun/16

A n o i l c a n i s t o b e m a d e i n f o r m o f a r i g h t c i r c u l a r c y l i n d e r t h a t b e

i n s c r i b e d i n a s p h e r e o f r a d i u s R . o b t a i n t h e m a x i m u m

v o l u m e o f t h e c a n .

Commented by sanusihammed last updated on 27/Jun/16

T h a n k s s o m u c h s i r .

Commented by Yozzii last updated on 27/Jun/16

V = π r^{2} h (V = v o l u m e o f c a n)

L e t θ = a n g l e s u b t e n d e d b e t w e e n t h e

h o r i z o n t a l a n d a l i n e j o i n i n g o n e p o i n t

o f c o n t a c t, b e t w e e n t h e c a n a n d t h e s p h e r e,

a n d t h e c e n t r e o f t h e s p h e r e .

s i n θ = \frac{h}{2 R} \Rightarrow h = 2 R s i n θ 0 < θ < 90 °

c o s θ = \frac{r}{R} \Rightarrow r = R c o s θ .

V = π R^{2} {c o s}^{2} θ \times 2 \times R s i n θ

V = π R^{3} c o s θ s i n 2 θ

V = \frac{π R^{3}}{2} (s i n 3 θ + s i n θ)

\frac{d V}{d θ} = π R^{3} (- s i n θ s i n 2 θ + 2 c o s θ c o s 2 θ)

W h e n \frac{d V}{d θ} = 0

\Rightarrow 2 c o s θ c o s 2 θ = s i n θ s i n 2 θ

t a n θ t a n 2 θ = 2

\frac{2 {t a n}^{2} θ}{1 - {t a n}^{2} θ} = 2

{t a n}^{2} θ = 1 - {t a n}^{2} θ

2 {t a n}^{2} θ = 1 \Rightarrow t a n θ = \frac{1}{\sqrt{2}} \Rightarrow θ = {t a n}^{- 1} \frac{1}{\sqrt{2}} 0 < θ < 90 °

\frac{d V}{d θ} = \frac{π R^{3}}{2} (3 c o s 3 θ + c o s θ)

\frac{d^{2} V}{d θ^{2}} = - \frac{π R^{3}}{2} (9 s i n 3 θ + s i n θ)

S u b s t i t u t i n g θ = {t a n}^{- 1} 2^{- 0.5} g i v e s \frac{d^{2} V}{d θ^{2}} < 0

\Rightarrow m a x (V) o c c u r s a t θ = {t a n}^{- 1} 2^{- 0.5} .

t a n θ = \frac{1}{\sqrt{2}} \Rightarrow c o s θ = \frac{\sqrt{2}}{\sqrt{3}} = \sqrt{\frac{2}{3}}

& s i n θ = \frac{1}{\sqrt{3}} .

∴ m a x (V) = 2 π R^{3} \times \frac{2}{3} \times \frac{1}{\sqrt{3}}

m a x (V) = \frac{4 π R^{3}}{3 \sqrt{3}} = \frac{1}{\sqrt{3}} V_{s p h e r e}

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