Calculating Volume of a Closed Cylinder with 600π Surface Area

[tweety1234]

Homework Statement

A closed cylinder has total surface area equal to $$600\pi$$.

Show that the volume, Vcm3, of this cylinder is given by the formula$$v = 300\pi-\pi r^3$$,
where r cm is the radius of the cylinder.
Find the maximum volume of such a cylinder.

Homework Equations

$$V= 2\pir^2 + 2\pirh$$

The Attempt at a Solution

not really sure where to start,

$$600\pi = 2\pir^2 + 2\pirh$$

how would I make this expression equal the other?

Thanks!

 

[HallsofIvy]

Homework Statement

A closed cylinder has total surface area equal to $$600\pi$$.

Show that the volume, Vcm3, of this cylinder is given by the formula$$v = 300\pi-\pi r^3$$,
where r cm is the radius of the cylinder.
Find the maximum volume of such a cylinder.

Homework Equations

$$V= 2\pir^2 + 2\pirh$$

A, the total surface area, is $2\pi r^2+ 2\pi rh$, not the volume.

The Attempt at a Solution

not really sure where to start,

$$600\pi = 2\pir^2 + 2\pirh$$

Yes, that is correct.

how would I make this expression equal the other?

Well, first you have to decide what the "other" expression is! What is the formula for volume of a cylinder? And you don't "make them equal". Solve the equation giving surface area for h and use that to replace h in the formula giving volume.

Thanks!

 

[tweety1234]

Latex gone horribly wrong.

this is the question;

A closed cylinder has total surface area equal to 600π. Show that the volume, Vcm3, of this cylinder is given by the formula V=300πr−πr3, where r cm is the radius of the cylinder.
Find the maximum volume of such a cylinder

n=pi

 

[tweety1234]

well the formula for the volume of a cylinder = v= 2\pir^2 + 2\pirh so if I set that equal to 600\pi, what do I do after that?

 

[tweety1234]

Well, first you have to decide what the "other" expression is! What is the formula for volume of a cylinder? And you don't "make them equal". Solve the equation giving surface area for h and use that to replace h in the formula giving volume.

so If I solve for 'h' from 600\pi = 2\pir^2 + 2\pirh I get; 300-r^2/r = h,

600\pi = 2\pir^2 + 2\pi(300-r^2)/r -----that does not simplify to the desired expression, what did I do wrong?

 

[llauren84]

so If I solve for 'h' from 600\pi = 2\pir^2 + 2\pirh I get; 300-r^2/r = h,

Actually h=(300-r^2)/r not 300-r^2/r

600\pi = 2\pir^2 + 2\pi(300-r^2)/r -----that does not simplify to the desired expression, what did I do wrong?

I don't even know which formula you are trying to substitute the h in. You need to put the h in the general formula for the area of a any cylinder.

 

[HallsofIvy]

so If I solve for 'h' from 600\pi = 2\pir^2 + 2\pirh I get; 300-r^2/r = h,

600\pi = 2\pir^2 + 2\pi(300-r^2)/r -----that does not simplify to the desired expression, what did I do wrong?

It looks like you have solve the area equation for h in terms of r and the replace h in the area equation! You want to replace h in the volume equation so as to get a formula for volume in terms of r only. The volume of a cylinder, of base radius r and height h, is $V= \pi r^2 h$.