" jointly proportional to $r$ and $h$" versus "directly proportional to both $r$ and $h$"

Question

Which of these statements correctly describe the formula $S = 2\pi rh$ ?

$S$ is jointly proportional to $r$ and $h$
$S$ is directly proportional to $r$ and $h$
$S$ varies directly with $r$ and $h.$

I am leaning towards choice 3, but am not sure.

This stuff on variation that might help ($k$ is a constant):

"S varies as x" means S = kx

"S varies jointly as x and y" means S = kxy

"S varies as x + y" means S = k(x + y)

"S varies inversely as x" means S = k/x.

How are the adverbs "jointly" and "directly" defined in your book? — J. M. ain't a mathematician, May 09 '11 at 15:28
I think directly means a constant is being multiplied with 1 variable, jointly means a constant is being multiplied with more than 1 variable. see edit. — Matt, May 09 '11 at 15:36
So, why did you choose the third as opposed to, say, the first? — J. M. ain't a mathematician, May 09 '11 at 15:36
Ah, because I didn't realize 2pi(r) wasn't a constant. 2pi would be k and radius and height would be variables so thinking about it that way, the first choice seems best. — Matt, May 09 '11 at 15:41
For what it's worth, I would use all three of those sentences to describe the relationship in question. I don't really see why your book is teaching you this. — Chris Taylor, May 09 '11 at 16:56

score 1 · Accepted Answer · answered May 12 '11 at 07:10

1

The verb "describe" in the question is in the plural. That would seem to imply that any number of answers can be selected. As Chris noted in the comments, all three sentences are OK (assuming an appropriate definition of the terms used).

answered May 12 '11 at 07:10

joriki

238,052

ryang · Answer 2 · 2023-06-25T14:07:19.563

Which of these statements correctly describe the formula $S = 2\pi rh$ ?

$S$ is jointly proportional to $r$ and $h$

This option ($S\propto rh$) directly describes the above formula, so is always correct.

$S$ is directly proportional to $r$ and $h$

$S$ varies directly with $r$ and $h$

These options ($S\propto r \quad\text{and}\quad S\propto h$) would be wrong if $r$ depends on $h.$

For example, the constraint $$h=3r$$ implies that $$6\pi r^2=S=\frac23\pi h^2,$$ which implies that $S$ is proportional neither to $r$ nor to $h.$

" jointly proportional to $r$ and $h$" versus "directly proportional to both $r$ and $h$"

2 Answers2