My Math Forum Find the Surface Area of an Open-topped Lid

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November 15th, 2011, 01:35 PM   #1
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Find the Surface Area of an Open-topped Lid

Find the surface area of this open-topped lid.

I know the formula for the surface area of a cylinder is 2pi(r^2)+2pi(rh), but I don't know how to calculate the inside of the lid.
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 November 15th, 2011, 02:10 PM #2 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus Re: Find the Surface Area of an Open-topped Lid [color=#000000]The total area is $\pi\cdot 100\cdot 500+\pi\cdot 99\cdot 500 +\pi\left(50^2-49.5^2)+\pi (50)^2+\pi (49.5)^2=104500\pi\;cm^2$.[/color]
 November 15th, 2011, 03:15 PM #3 Global Moderator   Joined: Dec 2006 Posts: 19,739 Thanks: 1810 That can't be right.
 November 15th, 2011, 05:43 PM #4 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,340 Thanks: 939 Re: Find the Surface Area of an Open-topped Lid Hard to tell...if you mean the outside plus the inside: bottom: 2(50^2)pi = 5000pi : (outside bottom) + (inside bottom + 1cm ring) outside wall: 2(50)(500)pi = 50000pi inside wall: 2(49)(500)pi = 49000pi Add 'em up: 104000pi ZardoZ used 99 as inside diameter: should be 98
November 15th, 2011, 07:23 PM   #5
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Re: Find the Surface Area of an Open-topped Lid

Hello, Pumpkin99

I wish you would explain this "open-topped lid" more clearly.

Quote:
 Find the surface area of this open-topped lid. I know the formula for the surface area of a cylinder is 2pi(r^2)+2pi(rh). but I don't know how to calculate the inside of the lid.

I would expect a "lid" to be like a pizza pan . . . flat bottom (or top), round, with short vertical sides.

You've given us a cylinder: 100 cm wide and 500 cm high, with a "wall" that is 1 cm thick.

You say "open-topped", so I assume it has a "floor" (with thickness 1 cm).

I assume that we want to paint this thing, so we want to know
[color=beige]. . [/color] the number of square centimeters of paint we will need.

The side view of this "lid" looks like this:

Code:
        1                   1
*---*       :       *---*
|   |       :       |   |
|   |       :       |   |
|   |       :       |   |
|   |499    :       |   |
500|   |       :       |   |
|   |       :  49   |   |
|   *-------+-------*   |
|           :    50     |
*-----------------------*

Outside area

On the top we have a "ring" with area:[color=beige] .[/color]$\pi(50^2)\,-\,\pi(49^2) \:=\:99\pi\text{ cm}^2$

On the "side", we have the curved area:[color=beige] .[/color]$2\pi(50)(500) \,=\,50,000\pi\text{ cm}^2$

On the bottom, we have a circle of radius 50:[color=beige] .[/color]$\pi(50^2) \,=\,2500\pi\text{ cm{^2$

Inside area

On the "side" we have the curved area:[color=beige] .[/color]$2\pi(49)(499) \,=\,48,902\pi\text{ cm{^2$

On the bottom, we have a circle of radius 49:[color=beige] .[/color]$\pi(49^2) \,=\,2401\pi\text{ cm}^2$

The total surface area is:[color=beige] .[/color]$103,902\pi\text{ cm}^2.$

 November 15th, 2011, 11:37 PM #6 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus Re: Find the Surface Area of an Open-topped Lid [color=#000000]Seems I had a different shape in mind! When you don't use pencil and paper and use only the mind this happens, seems I am not clever enough! [/color]
November 16th, 2011, 03:28 AM   #7
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Quote:
 Originally Posted by Pumpkin99 Find the surface area of this open-topped lid. I know the formula for the surface area of a cylinder is 2pi(r^2)+2pi(rh), but I don't know how to calculate the inside of the lid.
The question is somewhat unclear. The term "cylinder" is ambiguous anyway; it is so often used to refer to a right, circular cylinder that this usage can usually be assumed by default, but it still remains for the context to indicate whether it is a solid or a surface that is being referred to. The formula given above gives the (total) surface area of a solid (right, circular) cylinder. However, this doesn't clarify what is meant by "open-topped lid". What kind of lid is not "open-topped"? If the lid has a base, how thick is that base? If the above formula for the surface area of a cylinder is known and (presumably) understood, why isn't it obvious how to modify it slightly to give the surface area of the inside of the lid? The question seems to indicate that the sticking point in finding the surface area of the lid was finding the area of the inside of the lid; why would that be the case?

 November 16th, 2011, 08:02 AM #8 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,340 Thanks: 939 Re: Find the Surface Area of an Open-topped Lid Plus a "lid" is a "cover": "a removable or hinged cover for closing the opening, usually at the top, of a pot, jar, trunk, etc" So a lid with "walls" of length 500 cm sure seems strange; wonder if he means .5 cm

 Tags area, find, lid, opentopped, surface

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