Solve Optimization Problem for Max Cylinder Volume w/ 36cm Perimeter

[mattmannmf]

Just want to make sure I am doing it correct!

a Rectangle sheet of perimeter 36cm with dimensions x (vertical) and y (horizontal) is to be rolled into a cylinder where x= height and y= circumference. what values of x and y will give largest volume? Write volume in terms of only one variable "y" when solving.

So volume of cylinder:
V=2(pi)r*h
when applying our variables:
V= y * x
y= 2(pi)r
x=h

This I know is our primary equation: v=y*x for the cylinder

Now they gave us the perimeter which must be 36cm...so our 2nd equation will be the perimeter of the rectangle before its rolled up into a cylinder:
P=2x+2y
p=36

So i got (in terms of y):
36=2x+2y
18-y=x

Using this i applied what x equaled in our 2nd equation to our primary equation:
V=y* (18-y)

I then took the derivative and got
Dv/dt= 18-2y

I then solved for a critical point (making the derivative equal 0) and got y=9.

making my values to be
y=9cm
x=9cm

Does this seem right? Also would my domain be from (0,36)?

 

[phyzmatix]

Hi! I didn't actually check your whole solution since

So volume of cylinder:
V=2(pi)r*h

is wrong.

Also, did they specify which optimization method you should use? If not, you might want to check out the Method of Lagrange. I find it often easier to apply.

 

[Mark44]

I would advise you to continue with what you were doing, as it is a pretty straightforward way to go. As phyzmatix pointed out, you formula for the volume is incorrect. Your formula will give you the area (not volume) of the rectangular sheet, since you are just multiplying the length of the sheet (the circumference of the cylinder) by its height. The volume of the cylinder is the area of its base times its height.

 

[mattmannmf]

Ok, so the formula for Volume would be:

V= y^2 * x

y= 2 (pi) r
x= h

Alright, and as far as the domain its (0,36) correct?

 

[Mark44]

No, that's not the volume. The volume is the area of the circular base times the height. The base is a circle whose circumference is y. From this circumference you need to find the radius r, and then use it to find the area of the circle.

The domain for y is [0, 18], not (0, 36).

 

[mattmannmf]

ok, so after thinking it over the volume would actually be:

V= y^2/(4(pi)) * x
where y= 2(pi)r

 

[Mark44]

Yes, that's right. You know that x = 18 -y, so replace x in your equation for V to get V as a function of y alone. Then you can get V'(y) and find the value of y that maximizes the volume.