Surface area and volume of a cylinder

[scientist]

Could a tutor please check my work?

question:

What is the surface area and volume of a pressure vessel in the form of a cylinder with each end in the form of a hemisphere, if the overall length is 12 meters and the diameter is 3 meters.

solution:

given:
radius = 1.5m
diameter = 3m
height or length = 12m
pi=3.14

------------------

Total surface area of a cylinder = 2*pi*r*h + 2*pi*r^2

TSA = 2*3.14*1.5*12 + 2*3.14*1.5^2
= 113.04 m^2 + 14.13 m^2
= 127.17 m^2


Volume of a cylinder = pi* r^2*h

V= 3.14*1.5^2*12
= 84.8 m^3

Is this correct?

 

[Curious3141]

Overall length is 12m, and that includes the two radii of the hemispheric ends. So why are you assuming that the the height of the cylindrical portion is 12m?

Second problem : the surface area of the object is the surface area of the *wall* of the cylindrical portion *plus* the combined surface areas of the two hemispherical portions. You should not include the areas of the discs that make up a closed cylinder (like you did). And you did not consider the S.A. of the hemispherical ends at all.

Third problem : volume of the cylinder is certainly correct, but what about volumes of the hemispherical ends?

 

[scientist]

To: Curious3141

The height of the cylinder is 12 meters high. The length and the height are the same thing. Both are 12 meters.

The volume of the hemispherical ends = 2*pi*r^3 / 3.
V=2*3.14*1.5^3 / 3
= 21.195 / 3
= 7.065 m^3 at each end.

The total surface area of the hemispherical ends = 4*pi*r^2 / 2
TSA = 4*3.14*1.5^2 / 2
= 28.26 / 2
= 14.13 m^2 for both ends.

surface area of a cylinder = 2*pi*r*h + 2*pi*r^2
SA = 2*3.14*1.5*12 + 2*3.14*1.5^2
= 113.04 m^2 + 14.13 m^2
= 127.17 m^2

127.17 m^2 - 14.13 m^2 = 113.1 m^2 total surface area of the cylinder.

Can you check my work again?
From,
scientist

 

[HallsofIvy]

In your first post you said "the overall length is 12 meters" which would include the two hemispheres. Now you say "The height of the cylinder is 12 meters" which does not include the two hemispheres.

Which is it?

Why do you calclulate the volume of one hemisphere as \frac{2}{3}\pi r^3, then turn around a calculate the volume of both as \frac{4}{3}\pi r^3? They are both correct but you don't use the first one!

Curious3141 told you NOT to include the are of the ends of the cylinder- that is not part of the surface area of the pressure vessel. So you include it and then subtract it off?? (It's interesting that you get 3.04 for the lateral area of the cylinder, add the area of the two end circles, then immediately subtract off that area and get an answer of 3.1!)

Oh, and you never do calculate the whole surface area of the vessel.

The crucial point is whether the overall length is 12 m or only the length of the cylinder. Exactly what does your problem say?

 

[scientist]

Ok, let's get the height straightened out first. HallsofIvy asked:
The crucial point is whether the overall length is 12 m or only the length of the cylinder. Exactly what does your problem say? The problem in my book says the overall length is 12 meters. So there is only one number left for the height. And that is 3 meters, for the height.

scientist

 

[scientist]

VOLUME:

volume of the ends = 4*3.14*r^3 / 3 = 4*3.14*1.5 m^3 / 3 = 14.13 m^3
14.13m^3 / 2 = 7.065 m^3 at each end.

volume of cylinder = 3.14*r^2*h = 3.14*1.5^2*12m = 84.78m^3

volume total = 84.78m^3 - 7.065m^3 = 77.715m^3

-----------------------------------------

SURFACE AREA:

surface area of the ends = 4*3.14*1.5m^2 = 28.26m^2 / 2 = 14.13m^2 at each end.

surface area of the cylinder = 2*3.14*r^2 + 2*3.14*r*L = 2*3.14*1.5m^2 + 2*3.14*1.5m*12m = 127.17m^2

surface area total = 127.17 m^2 - 14.13 m^2 = 113.04 m^2

 

[HallsofIvy]

Ok, let's get the height straightened out first. HallsofIvy asked:
The crucial point is whether the overall length is 12 m or only the length of the cylinder. Exactly what does your problem say? The problem in my book says the overall length is 12 meters. So there is only one number left for the height. And that is 3 meters, for the height.

scientist

For the height of what? The hemispheres have diameter 3 m so the two of them add 3 m to the overall length. The length of the cylinder is 12- 3= 9 meters.

VOLUME:

volume of the ends = 4*3.14*r^3 / 3 = 4*3.14*1.5 m^3 / 3 = 14.13 m^3
14.13m^3 / 2 = 7.065 m^3 at each end.

Yes, that is correct. (Except that it is not "at each end"- that is the total volume of both ends.)

volume of cylinder = 3.14*r^2*h = 3.14*1.5^2*12m = 84.78m^3

NO! since the "overall length" is 12 m and the hemispheres account for 3 m of that, the length of the cylinder is 9 m, not 12! That's what we've been trying to tell you all along!

volume total = 84.78m^3 - 7.065m^3 = 77.715m^3

-----------------------------------------

SURFACE AREA:

surface area of the ends = 4*3.14*1.5m^2 = 28.26m^2 / 2 = 14.13m^2 at each end.

Well, not at "each end"- that's the total area of both ends.

surface area of the cylinder = 2*3.14*r^2 + 2*3.14*r*L = 2*3.14*1.5m^2 + 2*3.14*1.5m*12m = 127.17m^2

One more time and then I give up! The ends of the cylinder are NOT part of the surface- they contribute NO surface area! Oh, and the length of the cylinder is 9 m, not 12 m.

surface area total = 127.17 m^2 - 14.13 m^2 = 113.04 m^2

Why in the world are you subtracting the surface area of the hemispheres?

 

[scientist]

OK, the length is 9m NOT 12m. I understand that now. I have more of these type of questions. I will practice.