The Puzzle of the Cooper and the Vinter?

Question

A cooper and a vinter sat down for a talk, both being so groggy that neither could walk; Says cooper to the vinter, :IM the first of my trade, there's no kind of vessel but what I have made, And of any shape, sir, just what you will, and of any size, sir from a tun to a gill." " Then," says the cinter. "Your the man for me. " Make a vessel and the top and bottom are defined. The proportion is 15-9. 35 inches is the depth and can only hold 39 gallons.

Whats the dimensions, and show all of the work.

Answer 1

Ok, so it's not a realistic barrel. Sheesh. It's a math problem, for crying out loud. And, at its core, it's a conic section problem. What you're being asked is find the dimensions of a conic section with a given volume. To do that, you need to realize what a cone and a conic section actually are. A cone is just a right triangle spinning in a circle, and a conic section is a right trapezoid doing likewise. There's probably a formula for getting the volume of a conic section, but I don't know it, so I find it easiest to complete the cone (draw it out to its endpoint) get the volume of the full cone, get the volume of the cone I've added on, then subtract to get just the section I'm working with.

To start, draw the conic section as a right trapezoid. Mark the lengths you know: the short parallel side is 9x, the large parallel end is 15x, and the perpendicular side is 35. Then draw a vertical line down from the outside edge of the small end (the bottom of your mythical cask) to the large end, the length of which will also be 35. This turns your trapezoid into a rectangle and a right triangle, making it much easier to deal with:
_
I I\
I_I_\

Given that you know that you have a rectangle, the left segment of the base of the trapezoid (the bottom of the rectangle) must be 9x, which makes the base of the triangle 6x.

Now you'll put a right triangle on top of your rectangle to make the whole thing into one large right triangle. This triangle will be similar to (that is, the sides will be in proportion with) the triangle that you made in your trapezoid. You know that the base of your old triangle is 6x and the height is 35. Since the base of the new triangle is 9x, the height of the imaginary triangle must be (9/6)(35), or 52.5.

The volume of a cone is 1/3(pi)r^2h. (Remember that the base of the triangle is the radius of the circle.) You now have two cones: the imaginary one, and the bigger one which includes both the real and the imaginary. Figure out their volumes, and subtract out the imaginary:
Imaginary: 1/3(pi)(9x)^2 (52.5)
Big 'un: 1/3(pi)(9x + 6x)^2 (87.5)

Imaginary: 1417.5(pi)x^2
Big 'un: 6562.5(pi)x^2
Real: 5145(pi)x^2
(To make things easier from here on out, I'll use 3.14 for pi.
Real: 16155.3x^2 in^3

So we know what the volume of the section is in proportion to the radii of the top and bottom. Now we need to figure out how many cubic inches of fluid are in 39 gallons. Taking a quick trip to a conversion website, I find that there are almost exactly 231 cubic inches to a gallon, or 9009 in^3 in 39 gallons.

The rest is fairly simple:
16155.3x^2 = 9009
x^2 = .558
x = .747
9x = 6.721
15x = 11.201

You would normally express the size of a circular object in terms of its diameter (think pizzas or pies), so double those last two numbers to get a cask where the top is a disc of diameter 22.402 in, the bottom is a disc of 13.442 in, and the depth is 35 inches.

Whew! I'm exhausted!

Answer 2

Barrels aren't cylinders with straight sides--they curve out until they reach their widest point in the middle.

As far as dimensions, no clue.

How come the first part of the "puzzle" rhymes, and the second part sounds like a word problem from a math class? I don't think this is a riddle. I don't know how to do your homework without some paper, a writing implement, maybe a formula or something. Like I have that on Yahoo Answers.

I think you're going to have to do this on your own.