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A tennis ball can with radius r holds a certain number of tennis balls also with the same radius. The amount of space in the tennis ball can that is not occupied by the tennis balls equals at most the volume of one tennis ball. How many tennis balls does the can hold?
Barry sent me this problem months ago & I've been avoiding it because geometry scares me.
I finally shamed myself into attempting it just now & got an answer of 2. Unfortunately, I typed up my solution, loaded it to flickr, but flickr is on the blink so I can't post.
I solved it (assuming I did solve it) algebraically, then resorted to "logic and reasoning" to check.
Unfortunately, I'm confused by logic and reasoning at the moment.
I was thinking that because a sphere is 2/3 of a cylinder of same radius, with each ball you put inside a same-radius cylinder you end up with 1/3 of a ball's worth of empty space....which now implies to me that the answer should be 3 balls, not 2.
sigh
update (1):
I'm mixing things up
The 1/3 that's left over isn't 1/3 of a tennis ball. It's 1/3 of a cylinder with the same radius as the tennis ball.
I better forget the logic and reasoning & stick to algebra.
Assuming I didn't screw up the algebra, that is.
update (2):
OK, so in between dealing with screaming autistic youths, loading the dishwasher, & microwaving a taco for Jimmy, I realized that I don't need to know "how much of a tennis ball-sized volume is left over."
I just need to know how much empty volume is left over, period, then figure out how many multiples of that empty space add up to the volume of 1 tennis ball.
volume of tennis ball with radius r: 4/3πr^3
height of cylinder that fits just one tennis ball: 2r
volume of cylinder w/height of 2r: πr^2h = 2πr^3
vol. of cylinder - vol. of 1 tennis ball = vol. of empty space
2πr^3 - 4/3πr^3 = 2/3πr^3 empty space left over when 1 tennis ball is in cylinder
2 tennis balls leaves 2 empty spaces, each 2/3πr^3 in volume:
2/3πr^3 + 2/3πr^3 = 4/3πr^3, which is the volume of 1 tennis ball
so: 2 tennis balls
update (3):
Barry says the problem comes from Dolciani's Algebra 2! (I include the exclamation point because I'm happy to discover I am able to solve a problem from that book. cool.)
original wording:
A tennis ball can in the shape of a cylinder with a flat top and bottom of the same radius as the tennis balls is designed so that the space inside the can that is NOT occupied by the balls has volume at most equal to the volume of one ball. What is the largest nubmer of balls the can will contain?
20 comments:
I couldn't solve your problem because of the phrasing.
" A tennis ball can with radius r holds.."
I thought "can" was a verb. Every single time I read it. 6, 7 times I read it incorrectly. I never thought of a cylinder, until reading your equations. I just thought it was yet another unsolveable problem (how can a tennis ball of radius r hold other tennis balls of radius r? Couldn't they at least have said a basketball?), or a badly written problem where they meant that the basketball was radius r, and the tennis balls had the same radius as each other...
I'm an adult with a math degree. I wonder how many other people, especially children, get tripped up by word choice.
If the original problem had said " a can of tennis balls" or a "canister of tennis balls" I would have understood immediately.
I thought "can" was a verb. Every single time I read it.
Oh, that's awful!
Barry sent me this problem ages ago in the course of an email exchange (I've forgotten what we were talking about...I was probably whining about my inflexible knowledge...)
I can't remember, now, whether I had a context going in or whether I simply understood the wording.
I wish I did remember because I've noticed that math majors can get tripped up by language that doesn't trip up the rest of us & I'd like to understand the difference. I used to think that mathematicians were far more precise, but that's not it exactly.
Working with Temple I realize that I am also "precise" - or at least "hyperspecific"....but precision for me means that there is no such thing as a "true" synomym.
I wonder whether "math-types" are more precise about the structure of sentences while "writer-types" are more precise about the meanings of individual words???
In any case, on several different occasions I've noticed that word problems whose meaning seems obvious to me aren't clear for people who really know math.
Having blathered on about that, let me agree with the overall point---I've seen word problems in textbooks & on tests that were a mess for anyone no matter where your talents lie.
My apologies to Allison and others who had difficulty understanding the wording. I took the problem from Dolciani's Algebra 2 and shortened the problem.(I was in a hurry and didn't want to type the whole thing.) The original wording is: "A tennis ball can in the shape of a cylinder with a flat top and bottom of the same radius as the tennis balls is designed so that the space inside the can that is NOT occupied by the balls has volume at most equal to the volume of one ball. What is the largest nubmer of balls the can will contain?"
I took the problem from Dolciani's Algebra 2
wow!
you made my day!
I was thinking this was Algebra 1.
I had another Good Math Moment today....which I'll probably try to write a post about.
I'd gotten utterly confused by something & then managed to become un-confused in a couple of sessions instead of the couple of years it took me to figure out why invert-and-multiply is the way to divide a fraction by a fraction.
[I'd gotten utterly confused by something & then managed to become un-confused in a couple of sessions instead of the couple of years it took me to figure out why invert-and-multiply is the way to divide a fraction by a fraction.]
I love "un-confused."
It's better than still being confused, but on a higher level.
In my quest to make connections, I am wondering if invert-and-multiply doesn't have a parallel in the algebraic method of subtraction? The rule for integer subtraction is add the opposite (of the subtrahend) and for fraction division it is multiply by the reciprocal of the divisor. In both cases, we turn to an inverse operation, i.e. convert a subtraction problem into an addition problem and a division problem into a multiplication problem. So the analogue of ooposite integer would be reciprocal.
Just year-end musings. Probably confusion at a higher level.
hey!
were you the person who first wrote the comment about being confused at a higher level or was it Vlorbik??
I've always liked that
I've had too much champagne at the moment to process the invert-and-multiply part of your comment...
I'm sitting at my computer only because I have 3 1/2 hours left to use my $50-off coupon from Garnet Hill
so it's now or never
all of algebra, as far as I can tell, seems to be one big long protracted exercise in inversing
This reminds me of the sphere packing problem.
[were you the person who first wrote the comment about being confused at a higher level or was it Vlorbik??]
That was my contribution. I believe it is listed in KTM's W&W. I read the phrase somewhere eons ago and liked it instantly. It stuck in my mind ever since.
It's unfortunate that some clever phrases one runs into over time are forgotten. I remember reading a phrase during the cold war era that impressed me immensely for its cleverness and am surprised nobody picked up on it. I don't remember the author at this point. The phrase needs to be understood in the ideologically charged context of the time. It was an accusation that something was swallowed hook, party line and sinker. I think it could be adapted to educational matters.
I was just digging around ktm-1's Wit & Wisdom and laughed my head off when I came across this:
experts
I've quoted Steve and Ken so much I'm starting to sound like I know what I'm talking about.
- Susan S
And there is so much more there!
A little bit of googling revealed that the confusion quote is attributed to Enrico Fermi:
Before I came here I was confused about this subject. Having listened to your lecture I am still confused. But on a higher level.
I'd forgotten that line from Susan.
I'm going to have to post a wit and wisdom link.
A factoid I find interesting. I learned from Wikipedia that the word "algorithm" is derived from the name of the father of algebra:
Al-Khwārizmī, Persian astronomer and mathematician, wrote a treatise in Arabic in 825 AD, On Calculation with Hindu Numerals. (See algorism). It was translated into Latin in the 12th century as Algoritmi de numero Indorum,[1] which title was likely intended to mean "[Book by] Algoritmus on the numbers of the Indians", where "Algoritmi" was the translator's rendition of the author's name in the genitive case; but people misunderstanding the title treated Algoritmi as a Latin plural and this led to the word "algorithm" (Latin algorismus) coming to mean "calculation method". The intrusive "th" is most likely due to a false cognate with the Greek αριθμος (arithmos) meaning "number".
I had always assumed said father was an Arab. It turns out he was Persian.
oh that is interesting!
Back when I was first teaching C. I told him to remember algorithm as "Al Gore has no rhythm."
Or, alternatively, if you liked Al Gore you could say, "Al Gore has lots of rhythm."
Speaking of liking or not liking Al Gore, C. just came down and said Huckabee and Obama are the projected winners.
yikes
With education I think Obama may be OK -- but Huckabee -- aaaauuugggghhhh
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