Cone volume differentiation to find maximum value

Original post by frostyy

Question:

A cylinder has a radius r meters and a height h meters. The sum of the radius and the height is 2m.

a) Find an expression for the volume, V, cubic meters, of the cylinder in terms of r only.

b) Hence find the maximum volume. Leave your final answer in terms of pi. State clearly the value of r where this occurs.

I got this far and I'm not sure what to do:

EDIT: Indeed m is of course the unit.
The final line of your working is good. Use r+h=2, so r=2-h. Then differentiate it again (second derivative test!) and plug you r-values in. You want the value to be less than 0 such that your r-value is a maximum and NOT a minimum value!
Ultimately, use that value of r to find the volume!

(edited 8 years ago)

Reply 3

OP

I'm confused. Does this mean that I did part a) wrong (since I didn't take out the m)? I'm not really sure how to take both h and m out of the equation

Reply 4

Student403

19

Original post by frostyy

I'm confused. Does this mean that I did part a) wrong (since I didn't take out the m)? I'm not really sure how to take both h and m out of the equation

m is not a variable. It is just a measurement like cm. When you work with measurements you use the value. It's fine how it is. You literally just need to take it out of your final expresson as you have not manipulated the m

Reply 5

Original post by frostyy

I'm confused. Does this mean that I did part a) wrong (since I didn't take out the m)? I'm not really sure how to take both h and m out of the equation

V=pi*r^2*h

r+h=2
so, h=2-r. *plg this into the above*

now, V=pi*r^2*(2-r); this is the answer to a).

Then differentiate, use your method up until that point. Use second derivative test to determine that it's a maximum and finally find the maximum volume!

Reply 6

OP

ok, so I got to f''(x) = 4pi - 6pi*r > 0

-6pi*r < -4pi
r < (-4pi / -6pi) [should I not flip the sign around here?)
r < 2pi / 3pi

am I going the right direction? what do I do know? Do I literally just plug 2pi / 3pi into v = pi*r^2(2 - r) ?

Reply 7

Original post by frostyy

ok, so I got to f''(x) = 4pi - 6pi*r > 0

-6pi*r < -4pi
r < (-4pi / -6pi) [should I not flip the sign around here?)
r < 2pi / 3pi

am I going the right direction? what do I do know? Do I literally just plug 2pi / 3pi into v = pi*r^2(2 - r) ?

Qué!?!??

v = pi*r^2(2 - r) is the function!
v' = -pi*r*(3r-4)
find the zeros: r=0, r=4/3
v''=-2*pi*(3r-4)
plonk in 0 and 4/3
0 yields a positive value, so minimum, throw out.
4/3 yields negative answer, so maximum. This is the value: r=4/3.
Now, use V = pi*r^2(2 - r) to find the max vol!!

Reply 8

OP

Original post by Bath_Student

Qué!?!??

v = pi*r^2(2 - r) is the function!
v' = -pi*r*(3r-4)
find the zeros: r=0, r=4/3
v''=-2*pi*(3r-4)
plonk in 0 and 4/3
0 yields a positive value, so minimum, throw out.
4/3 yields negative answer, so maximum. This is the value: r=4/3.
Now, use V = pi*r^2(2 - r) to find the max vol!!

I dont understand. So I dont even use the 2pi*r^2 - pi*r^3 that I got from expanding the brackets of pi*r^2(2 - r)?

Reply 9

Original post by frostyy

I dont understand. So I dont even use the 2pi*r^2 - pi*r^3 that I got from expanding the brackets of pi*r^2(2 - r)?

You do, then you differentiate.

v=2pi*r^2 - pi*r^3
v'=4pi*r - 3pi*r^2
=> v'=pi*r (4-3r)=-pi*r*(3r-4) [what I got above!].

Reply 10

OP

Original post by Bath_Student

You do, then you differentiate.

v=2pi*r^2 - pi*r^3
v'=4pi*r - 3pi*r^2
=> v'=pi*r (4-3r)=-pi*r*(3r-4) [what I got above!].

sorry that I'm asking so many questions, but I'm not sure what you did in the 2nd differentiation.

How did you get from pi*r(4-3r) to -2pi*(4-3r)?

Reply 11

Apolexian

12

Original post by frostyy

Question:

A cylinder has a radius r meters and a height h meters. The sum of the radius and the height is 2m.

a) Find an expression for the volume, V, cubic meters, of the cylinder in terms of r only.

b) Hence find the maximum volume. Leave your final answer in terms of pi. State clearly the value of r where this occurs.

I got this far and I'm not sure what to do:

2m is 2 meters

v=pir^2(2-r)
find dv/dr
set dv/dr=0
find value of r
put back into v=pir^2(2-r)
find v

Reply 12

OP

Original post by Apolexian

2m is 2 meters

v=pir^2(2-r)
find dv/dr
set dv/dr=0
find value of r
put back into v=pir^2(2-r)
find v

no problem with finding a v :u:

anyway so I don't even have to do the second differentiation?

Reply 13

Apolexian

12

Original post by frostyy

no problem with finding a v :u:

anyway so I don't even have to do the second differentiation?

no

Reply 14

OP

Original post by Apolexian

no

safe g

Reply 15

Original post by Apolexian

no

keep quiet if you're don't know what you're on about.

Original post by frostyy

-

You absolutely HAVE to do the second derivative test, because TWO values of r solved the above function, but only ONE yields a maximum! We were asked to find the maximum volume, NOT the minimum.

Have a read: http://mathworld.wolfram.com/SecondDerivativeTest.html

Reply 16

Original post by frostyy

no problem with finding a v :u:

anyway so I don't even have to do the second differentiation?

If only we could swap skills.. I could do with a v :smile:

Reply 17

OP

Original post by Bath_Student

If only we could swap skills.. I could do with a v :smile:

I guess we all want what we don't have :colondollar:

after I find the second derivative, do I have to do with it? I mean, the value of r (4/3) doesn't change, so the results isn't affected?

Reply 18

Original post by frostyy

I guess we all want what we don't have :colondollar:

after I find the second derivative, do I have to do with it? I mean, the value of r (4/3) doesn't change, so the results isn't affected?

Tell me your v-finding skills now. I've invested a good 15 mins into this trivial problem.

Anyway, do you appreciate that two values of r solve the above problem? If so, tell me which one to use, because I forgot. I am clueless. Why should one be better than the other??????!?!?!?!??!

The answer is that one yields a minimum value, and one a MAXIMUM. Entiendes, tío?

Reply 19