Question

A hollow cylinder, open at one end, is constructed of thin sheet metal. The total external surface area of the cylinder is 192pi cm^2. The cylinder has a radius of r cm and a height of h cm.

(i) express h in terms of r and show that the volume, Vcm^3 of the cylinder is given by
V=pi/2(192r-r^3).

Answer 1

i give up... sam's latex speed is even faster than my typing!

Answer 2

Thats area
\[A=\pi r^{2}+2\pi r h=192\pi\]

Answer 3

Yes, I thought so.. I was just about to ask :)

Answer 4

Find h from
\[A=πr^{2}+2πrh=192π\]

Answer 5

Oh. Ok.

Answer 6

192pi-pi r^2 over 2 pi r =h

Answer 7

h should be
\[h=\frac{192-r^{2}}{2r}\]

Answer 8

Yes

Answer 9

\[V = πr^{2}h\]
\[V = πr^{2}\frac{192-r^{2}}{2r}\]

Answer 10

Ok. Thanks. There's a second part of the question. Should I post it here?

Answer 11

post here

Answer 12

Given that r can vary,

(ii) find the value of r for which V has a stationary value.

Answer 13

I mean V=0?

Answer 14

Is that what I have to do?

Answer 15

you need to differentiate V then set dV/dr=0

Answer 16

So, how do I do that?

Answer 17

\[V=πr^{2}h\]
\[\frac{dV}{dr}=~~~96\pi-\frac{3\pi}{2}r^{2}~~=0~\]
\[96\pi-\frac{3\pi}{2}r^{2}~~=0\]
\[96\pi=\frac{3\pi}{2}r^{2}~~\]
\[\frac{3}{2}r^{2}=96\]
\[r^{2}=64\]
r=8

Answer 18

Wait, when you differentiate 96pi, doesn't that equal 0?

Answer 19

\[V=\frac{\pi}{2}(192r-r^3)\]
\[V=96\pi r-\frac{\pi r^{3}}{2}\]
\[\frac{dV}{dr}=96\pi -\frac{3\pi r^{2}}{2}\]

Answer 20

Oh, Ok. Silly me!

Answer 21

The last part of the question says

find this stationary value and determine whether it is maximum or minimum?
Do I differentiate it again?

Answer 22

r=8, substitute into
\[V=96\pi r- \frac{\pi r^{3}}{2}\]
\[V=512\pi\]

Answer 23

for max or min, differentiate again

Answer 24

would it be -3pi r
= -3 pi times 8
= <0
= maximum point?

Answer 25

yes

Answer 26

Thank you :)!