What is the largest possible volume for this cylinder?

[abizan]

Homework Statement

A rectangle with a perimeter of 40 cm is rotated around one of its sides creating a right cylinder. What is the largest possible volume for this cylinder?

Homework Equations

Volume of a cylinder= pi*r^2*h
Perimeter of a rectangle= 2x + 2L

The Attempt at a Solution

I know one of my equations is
40=2x+2h
And then I isolated h by itself and got:
h= 40-2x / 2
But for my volume equation I have two variables on one side(r & h) and that's where I'm stuck. I don't understand how the perimeter of the rectangle can relate to the volume of the cylinder. How am I suppose to write x in terms of radius? The answer on the back of my textbook says 3723.37cm^3 for what its worth.

 

[fresh_42]

How am I suppose to write x in terms of radius?

Draw a picture. How is the radius related to the rectangle?

 

[abizan]

Draw a picture. How is the radius related to the rectangle?

Is the circumference of the circles on the cylinder equal to the lengths of the rectangle? Because I tried that and I did not get the right answer...

 

[LCKurtz]

Homework Statement

A rectangle with a perimeter of 40 cm is rotated around one of its sides creating a right cylinder. What is the largest possible volume for this cylinder?

Homework Equations

Volume of a cylinder= pi*r^2*h
Perimeter of a rectangle= 2x + 2L

The Attempt at a Solution

I know one of my equations is
40=2x+2h
And then I isolated h by itself and got:
h= 40-2x / 2
But for my volume equation I have two variables on one side(r & h) and that's where I'm stuck. I don't understand how the perimeter of the rectangle can relate to the volume of the cylinder. How am I suppose to write x in terms of radius? The answer on the back of my textbook says 3723.37cm^3 for what its worth.

You have used L and h for the same thing. You have one side of the rectangle is x and the other is (40-2x)/2 = 20-x. (Note the necessary parentheses). What is the volume if that rectangle with sides x and 20-x is rotated about an edge?

 

[fresh_42]

Imagine a rectangle, say with a long and a short side. Then fix a long stick along the long side. This gives you a flag. Wave it so it circles around your stick. So radius, height and rectangle sides are actually only 2 lengths. Using the perimeter allows you to reduce it to only one left: x.

 

[Alexiy]

Here you see the direct relationship. You have that 40=2(r+h) directly is your 40=2(X+L)=2(r+h).You can express h=20-r. Plug this into your equation of volume, you will get a function V which depends on r, V(r). Now find the maxima, or if you like, the derivative of V with respect to r, that's the slope of the function, now maximum will be if the slope is zero. So $\frac {dV}{dr} = 0$ You will get an quadratic equation with two solutions, one of them will make sense, and when you get the $r_{max}$ plug that into the volume equation and you will get 3723,37. Try it.

 

[abizan]

Here you see the direct relationship. You have that 40=2(r+h) directly is your 40=2(X+L)=2(r+h).You can express h=20-r. Plug this into your equation of volume, you will get a function V which depends on r, V(r). Now find the maxima, or if you like, the derivative of V with respect to r, that's the slope of the function, now maximum will be if the slope is zero. So $\frac {dV}{dr} = 0$ You will get an quadratic equation with two solutions, one of them will make sense, and when you get the $r_{max}$ plug that into the volume equation and you will get 3723,37. Try it.

hmm okay I get this but I'm confused as to why the rectangle is only half of the cylinder? If a rectangle forms the cylinder, shouldn't the width of the rectangle be the diameter of the cylinder rather than the radius?

 

[Alexiy]

hmm okay I get this but I'm confused as to why the rectangle is only half of the cylinder? If a rectangle forms the cylinder, shouldn't the width of the rectangle be the diameter of the cylinder rather than the radius?

Because that's what it says in the text of the problem. The problems states that the rectangle rotates on one of its sides? here's a picture:

 

[abizan]

Because that's what it says in the text of the problem. The problems states that the rectangle rotates on one of its sides? here's a picture:

wow I completely missed that. Thank you so much!

 

[abizan]

Imagine a rectangle, say with a long and a short side. Then fix a long stick along the long side. This gives you a flag. Wave it so it circles around your stick. So radius, height and rectangle sides are actually only 2 lengths. Using the perimeter allows you to reduce it to only one left: x.

Oh I get it now so the radius is one of the rectangle's sides. Thanks for the help!