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C2 Differentation help please!

Hi, I would appreciate if somone can help me out with this question that I can't solve :frown:

r is the radius..

A closed cylinder has total surface area equal to 600(pi). Show that the volume, Vcm(cubed), of this cylinder is given by the formula V=300(pi)r - (pi)r(cubed), where r cm is he radius of the cylinder. Find the maximum volume of such a cylinder. :confused:

No idea where to start.

Thanks..

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Reply 1

zCtpts2T

Is your surface area right? I would of thought it would be something like:

S = 600 \pi r

where r is the radius of the cylinder (because S is dependent on r:

dS = rd\theta dz

)

Edit: Ignore this

Reply 2

Glutamic Acid

Using the formula for the surface area of a cylinder: The area of the circles at the end + the middle bit, you can say

2\pi r^2 + 2\pi rh = 600\pi

. Start off by dividing by 2pi to make things simpler.

Using the formula for the volume of a cylinder, you can say

V = \pi r^2h

. Rearrange the second equation to get it in terms of h, then substitute this in place of h in the first equation. Rearrange and tidy up.

Reply 3

silent ninja

Beat me to it.

To find the maximum volume, it's got something to do with differentiation :p:

Reply 4

kinglrb

Glutamic Acid

Using the formula for the surface area of a cylinder: The area of the circles at the end + the middle bit, you can say

2\pi r^2 + 2\pi rh = 600\pi

. Start off by dividing by 2pi to make things simpler.

Using the formula for the volume of a cylinder, you can say

V = \pi r^2h

. Rearrange the second equation to get it in terms of h, then substitute this in place of h in the first equation. Rearrange and tidy up.

Okay sounds good, but I am a bit dazzled, if you don't mind please bear with me and explain. Sorry :frown:

Appreciated :wink:

Reply 5

silent ninja

let area= blah blah like Glutamic has posted. Since you dont know the length of the cylinder, let the length be h (or l, or whatever letter you like). Rearrange this equation in the form h=...

Second equation, define the volume in terms of pi, r and h. Then substitue h from the above.

Reply 6

kinglrb

Glutamic Acid

Using the formula for the surface area of a cylinder: The area of the circles at the end + the middle bit, you can say

2\pi r^2 + 2\pi rh = 600\pi

. Start off by dividing by 2pi to make things simpler.

Using the formula for the volume of a cylinder, you can say

V = \pi r^2h

. Rearrange the second equation to get it in terms of h, then substitute this in place of h in the first equation. Rearrange and tidy up.

Okay so I get the first but now, because 2 pi r 2 + 2 pi r h is the surface area that must = 600 pi

Reply 7

Glutamic Acid

kinglrb

Okay sounds good, but I am a bit dazzled, if you don't mind please bear with me and explain. Sorry :frown:

Appreciated :wink:

No problem. The surface area of a cylinder is the two ends of the cylinder, plus the bit joining them. The bits at the end area circles, so you can use pi*r^2 to find the area. And there are 2, so it's 2pi*r^2. As for the bit in the middle, if you were to roll it out it'd be a rectangle. The width is the circumference of the circle, the length is the height, so it's 2pi*r*h.

As for the area, that's the area of the circle * height. So it's pi*r^2*h.

Reply 8

Glutamic Acid

kinglrb

Okay so I get the first but now, because 2 pi r 2 + 2 pi r h is the surface area that must = 600 pi

Yep. (And you have a common term of 2pi from each term which you can cancel.)

Reply 9

kinglrb

what is 600 pi divided by 2pi? is it 300 pi? Ignore that my bad

Reply 10

Glutamic Acid

For maximizing the volume, you need to differentiate

V = 300\pi r - r^3 \pi

with respect to r. Set the derivative equal to zero, solve, and then substitute back into the original formula. (And use the second derivative if you want to prove it)