where does this (1/2) come from

allegansveritatem

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The following is from a precalculus text I am studying by Earl Swokowski. The cylinder being referenced has a radius of 2 in. First he gives the formulas for volume of a cylinder and for volume of a cone and right after each formula he gives what we get when we are working with a cylinder and cone of 2in radius.
11498

What I want to know is what is that 1/2 doing there next to the h in the second line after the second equals sign? I mean, I don't see it indicated in the formula to right.
 

ksdhart2

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With only this brief snippet of information, I reach the same conclusion that you do - the 1/2 shouldn't be there. However, because the author specifically denotes \(h_{cylinder}\) and \(h_{cone}\), I can make an educated guess that \(h\) all by itself has a different meaning and there's probably information about its meaning in the section preceding these formulas. To me, it appears that the height of the cylinder is \(h\) and the height of the cone is half this value \(\left( \frac{1}{2}h \right) \)
 

allegansveritatem

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what you say is probably right. I don't have access to the book right now so I have to wait til tomorrow to check it. The problem has to do with finding the total volume of a cylinder with a cone on the end of it that is being used as a hopper to fill grain trucks, if I am correctly remembering. I don't recall there being any specification as to the ratio of the length of the cylinder to the length of the cone. I will post the whole problem tomorrow. Thanks for the reply.
 

Jomo

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I too don't see why the 1/2 is there but it is certainly included after the last equal sign. (1/3)(22)(1/2) = (1/3)(4)(1/2) =2/3
 

Denis

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MarkFL

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It seems reasonable to me that the height of the conical section of the silo is half that of the cylindrical portion, which is likely stated somewhere.

 

Denis

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Very droll Jean-Marc !
 

allegansveritatem

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The following is from a precalculus text I am studying by Earl Swokowski. The cylinder being referenced has a radius of 2 in. First he gives the formulas for volume of a cylinder and for volume of a cone and right after each formula he gives what we get when we are working with a cylinder and cone of 2in radius.
View attachment 1149811509

What I want to know is what is that 1/2 doing there next to the h in the second line after the second equals sign? I mean, I don't see it indicated in the formula to right.
yep, you got it. I should read things more closely. Here is the problem:
 

allegansveritatem

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What KS said is not "probably right": it is "definitely right"!.

You're wasting your time on a typo...
well, I didn't really think the book was wrong. But I just couldn't figure out how it was right. If I had the brains I was born with I would have checked the problem again.
 

Denis

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As Ralph Kramden would yell: Bang Zoom !!
 
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