cipker
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cipker 發問於 科學及數學數學 · 1 十年前

# AM>=GM

Show that the surface area S, of a closed cylinder of volume V, with base radius r is:

S = 2(pi)r^2 +2V/r

Using AM>= GM only,

Show that:

S/[6(pi)] >= [V^2 / 4(pi)^2]^(1/3)

If the volume is fixed, what is the ratio of the base radius r to the height h when the surface area is a minimum?

### 1 個解答

• 1 十年前
最愛解答

S = 2πr^2 + 2πrh

S = 2πr^2 + 2 (πr^2)h / r

S = 2πr^2 + 2 V / r

;;;;;;

S / (6π) >= ∛ ( V^2 / 4π^2 )

[2πr^2 + 2 (πr^2)h / r] / (6π) >= ∛ [ (πr^2 h)^2 / 4π^2) ]

( r^2 + rh ) / 3 >= ∛ [ (r^4) (h^2) / 4 ]

LHS

= ( r^2 + rh/2 + rh/2 ) / 3

>= ∛ [(r^2)(rh/2)(rh/2)]

= ∛ [ (r^4) (h^2) / 4 ]

= RHS

2010-07-23 20:07:57 補充：

If the volume is fixed ,

r^2 = rh/2 = rh/2

r = h/2

r : h = 1 : 2

2010-07-23 20:17:14 補充：

S = 2πr^2 + 2πrh

S = 2πr^2 + 2 (πr^2)h / r

S = 2πr^2 + 2 V / r

;;;;;;

S / (6π) >= ∛ ( V^2 / 4π^2 )

[2πr^2 + 2 (πr^2)h / r] / (6π) >= ∛ [ (πr^2 h)^2 / 4π^2) ]

( r^2 + rh ) / 3 >= ∛ [ (r^4) (h^2) / 4 ]

LHS

= ( r^2 + rh/2 + rh/2 ) / 3

>= ∛ [(r^2)(rh/2)(rh/2)]

= ∛ [ (r^4) (h^2) / 4 ]

= RHS

;;;;;;;;

' = ' holds if and only if r^2 = rh/2

r : h = 1 : 2 when the surface area is a minimum.

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