點計capacity of a cylinder

Capacity of cylinder=
Let radius be r

π r^2 x height of cylinder= Capacity of cylinder

π r^2= base area of cylinder
base area of cylinder x height = capacity of cylinder

2007-06-05 19:24:33 補充：
I don't know what answer one is about...I thought the question is how to calculate the capacity of cylinder...Why talk about hemisphere?! o

capacity of hemispherical part = 207π cm3 (2/23) = 18π cm3

volume of hemispherical part = (2/3)πr3 = 18π cm3

r = 3 cm

radius = 3 cm

PART B

capacity of cylidrical part = 207π cm3 (21/23) = 189π cm3

let the height be h

πr2h = 189π cm3

h = 21 cm

add the radius of hemisperical part together, we have

height = 21 + 3 = 24 cm

a)

Volume of the right circular cylinder,
πr² = 48π
∴h = 48/r²

Surface of the cylinder
= S
= 2πr² + 2πrh
= 2πr² + 2πr(48/r²)
= 2πr² + 96π/r

b)

The rate of change of S with respect to r
= dS/dr
= 2π(2r) - 96π/r²
= 4πr(1 - 24/r³)

When dS/dr = 0,
4πr(1 - 24/r³) = 0
∴1 = 24/r²
∴r² = 24
∴r = 2√3 ......(r＞0)

Considering
when r = 2√3 - Δr,dS/dr＜0
when r = 2√3 + Δr,dS/dr＞0

∴S attatins a minimum value at r = 2√3 cm, or 3.5 cm (corr to 1 d.p.)

b) 頭兩段要更正，幾處 r² 應是 r³ ，答案就啱：

The rate of change of S with respect to r
= dS/dr
= 2π(2r) - 96π/r²
= 4πr - 96π/r²
= 4π(r³ - 24)/r²

When dS/dr = 0,
4π(r³ - 24)/r² = 0
∴r³ - 24 = 0(∵r≠0)
∴r³ = 24
∴r = 2√3 ......(∵r＞0)