Help with a possible math discrepancy?

Factual Questions
1

I am attempting to resolve an apparent mathematical discrepancy I encountered while helping one of my kids with her homework.

The problem is as follows:

Given a cone with a base radius r of 5 cm and a slant height (L) of 12 cm, where the surface area in square cm equals the volume in cubic cm, solve for the height (h) of the cone. The formulas are helpfully provided:

Surface area = pi x r^2 + pi x rL

Volume = 1/3 x h x pi x r^2

So we know we can set these two formulas as an equation:

pi x r^2 + pi x rL = 1/3 x h x pi x r^2

Substituting the known values:

pi(5)^2 + pi(5)(12) = 1/3 x h x pi(5)^2

Which then gives:

25pi + 60pi = 1/3 x h x 25pi

Simplifying by combining terms and dividing both sides by pi gives:

85 = 1/3 x 25 x h

255 = 25 x h

10.2 = h

However, it occurred to me that one could calculate the height (h) given the base radius ® and slant height (L) using the Pythagorean theorem, since in a right circular cone r^2 + h^2 should equal L^2, correct?

However, if we do this, we get 25 + h^2 = 144

h^2 = 119

Which gives h = 10.9087…

So I feel like I must have either made a mistake or a wrong assumption somewhere, but I can’t figure it out. Can anyone help me on this?

2

I don’t see how you got those two lengths. The requirement that the surface area numerically equals the volume already contradicts that, as you observe.

3

I don’t see any errors on your part. The contradiction indicates that given the radius and slant length, the surface area does not equal the volume.

4

If r = 5 then your equation implies h = 75/8 , no? Either way, you do not then obtain L = 12

5

Those two measurements were provided in the text of the problem itself.

7

Correct; using the calculated value of h from the first equation does not yield L = 12 by the Pythagorean method either.

So I am assuming that the consensus answer is that a right circular cone cannot simultaneously have (r = 5, L = 12) and numerically equal surface area and volumes, correct?

8

Right.

9

Thanks to everyone for the help! :slightly_smiling_face:

10

There are a lot of bad worksheets like this floating around, where too much information is given, and the information given is contradictory. If the student solves it using exactly the same method that the question-writer expects, then they’ll get the same answer. But if they use any other method, they’ll get a different answer. And then they’re likely to conclude that math is stupid, and the world loses a promising young mathematician, instead of concluding that the person who wrote the problem is stupid.

11

See also this current thread

and my post #7.

12

I suppose a more interesting problem might be: in a right circular cone, for what value(s) of r, h, and L does the surface area numerically equal the volume? But that’s probably beyond the scope of the class in question.

13

Much more likely: The teacher insists, without further consideration, that the answer in the book is right and the kid is wrong (and stupid, if he/she tries to argue) and the world loses a promising young mathematician because the stupid teacher flunked the same out way back in 7th grade.

14

Or, even more likely, the teacher is a reasonable person and works with the kid to understand and explain why the problem as written doesn’t make sense, but then the teacher gets lumped in with the few bad apples that outweigh the good math teachers in the world.

15

A teacher who’s really a math teacher will probably do as @yearofglad says. A teacher who’s unqualified to teach math but is forced to do so by how the district structures their job would pull a @Senegoid.

16

Yes.

I prefer to use open-source textbooks, when I am given the option, as I am strongly of the opinion that the textbook industry is a giant racket, and do not want my students having to buy into their bullshit. But with an open-source book, you get what you pay for.

I taught College Algebra over the summer, and there were several instances where a student told me they’d done a problem several times, and kept coming up with the same answer, and couldn’t understand why they weren’t getting the answer in the “back” of the book. So I would do the problem myself, and lo and behold, the student was right and the book was wrong.

I actually think this is good in a way, providing a teaching moment. Much like with the OP. It shows that we shouldn’t just blindly accept that what we’re told is correct. I would go over the offending problem in the next class meeting, and make the point that students should always question.

17

Copying an exercise problem from somewhere when compiling a problem set is totally OK, but if it turns out the problem does not make sense, or, heavens help you, you cannot solve it yourself, that’s on you :slight_smile:

18

It is perfectly clear that the problem is nonsense. But there is a more serious question here. To equate an area and a volume is really a category error.

19

I disagree. It may be a little contrived and/or pointless, as in, “When would you ever want a cone where the surface area and volume are equal, ignoring units,” but the question is pretty clear - the surface area in square centimeters and the volume in cubic centimeters.

If I ask, “What is the volume of a cube in cubic centimeters?”, an answer of 64, for example, is perfectly fine. I’ve already given you the units - the part of the answer I’m asking for is just a number.

(I would’ve used the language “ignoring units” but I do think it’s fine as written.)

20

First the question is, as the OP realized, internally contradictory and therefore nonesense. And I stick to my point about comparing volumes and areas. Yes, you could ask for the dimensions of a cone for which the actual numerical value of volume and area are the same and you might even solve it. But it is not inherently of any interest.

21

It’s similar to problems where one angle is labeled as 3x+7 degrees and another is labeled as 5x+3 degrees, and you’re supposed to see that they add up to a right angle, and so you can solve that x = 10. Yes, you can do it, and yes, it’s a problem that combines algebra and geometry, like the standards require, but it’s a stupid problem, because why would you ever be given the angles that way?