Phil Quiz for Wednesday

[Philip Coates]

A rectangular open-topped metal tank is to hold 256 cubic feet of liquid. What are the dimensions of the tank that require the least material to build?

Show your steps and explain your reasoning.

Edited August 18, 2010 by Philip Coates

[Brant Gaede]

A rectangular metal tank is to hold 256 cubic feet of liquid. What are the dimensions of the tank that require the least material to build?

Show your steps and explain your reasoning.

I'd guess that since the compression of any side would destroy, not create, volume it'd have to be a perfect cube. Imagine a box an inch wide--the sides would have to go on and on an incredible distance.

--Brant

[merjet]

Yes, a cube almost 6.3 feet on each side.

It's easily shown with differential calculus that for a rectangle with a given perimeter, the maximum area is achieved with a square (length = width). That insight can be extrapolated to 3 dimensions and could even be proven with a little more complicated calculus.

[Brant Gaede]

I'm not competent to explore or express this mathematically. Phil could have made it a little more complicated and left me adrift at sea.

--Brant

[9thdoctor]

Why wouldn't it be the cube root of 256? It seems so easy that there must be a trick, but I don't see it. You learn this in Junior High, right?

Edited August 18, 2010 by Ninth Doctor

[Brant Gaede]

Why wouldn't it be the cube root of 256? It seems so easy that there must be a trick, but I don't see it. You learn this in Junior High, right?

And forget it.

--Brant

[Brant Gaede]

Maybe Phil meant this for somewhere else.

--Brant

[merjet]

Why wouldn't it be the cube root of 256? It seems so easy that there must be a trick, but I don't see it. You learn this in Junior High, right?

Yes, and the cube root of 256 is almost 6.35. I misread or misremembered the 256 and used 250 instead, which is why I said 6.3.

[Philip Coates]

> I'd guess that since the compression of any side would destroy, not create, volume it'd have to be a perfect cube. Imagine a box an inch wide--the sides would have to go on and on an incredible distance. [brant]

> Yes, a cube almost 6.3 feet on each side. It's easily shown with differential calculus that for a rectangle with a given perimeter, the maximum area is achieved with a square (length = width). That insight can be extrapolated to 3 dimensions [Merlin]

> Why wouldn't it be the cube root of 256? It seems so easy that there must be a trick, but I don't see it. You learn this in Junior High, right? [ND]

> the cube root of 256 is almost 6.35 [amended/Merlin]

Guys, that is not correct.

The tank is not a cube and it's not the cube root. Nor can you extrapolate from the two dimensional case. Plus it's not maximum area but minimum surface area. Nor did you learn this in Junior High. (Brant, we're not talking about minimizing volume but surface area.)

I just realized this problem, while it looks simple, is actually too hard for this board. Probably no one here except perhaps Baal or Dragonfly will know how to do this.

So: I'll leave the problem up for them to try if they know how. [And then I'll give the answer.] Meanwhile, I'll post a modified, but considerably simpler problem as "Phil Quiz for Wednesday -- Amended."

Edited August 18, 2010 by Philip Coates

[Brant Gaede]

I thought a cube was also a rectangle.

--Brant

[merjet]

Guys, that is not correct.

The tank is not a cube and it's not the cube root. Nor can you extrapolate from the two dimensional case. Plus it's not maximum area but minimum surface area. Nor did you learn this in Junior High. (Brant, we're not talking about minimizing volume but surface area.)

First, my answer did assume minimum surface area. You didn't say whether or not the tank had a top. I assumed it did. Moreover, all you said about the shape was it was "rectangular", which is ambiguous when talking about 3 dimensions.

[Brant Gaede]

Some tanks don't have tops but that'd made the quiz a gottcha thing and I don't think Phil was trying to do that.

--Brant

somehow I suspect the answer won't be very interesting

[tjohnson]

A rectangular metal tank is to hold 256 cubic feet of liquid. What are the dimensions of the tank that require the least material to build?

Show your steps and explain your reasoning.

I don't think you have enough information to calculate that. If we knew the sides were 2 ft high, for example, then you would have the surface area, S=xy+4x+4y and 2xy=256. Then y=128/x and substituting you get S=4x + 496/x + 128. Taking the derivative, dS/dx = 4 - 496/x^2 and setting to 0 yields that x=sqrt(128), or around 11.31 ft. Then y=11.31 also.

[Philip Coates]

Sorry, tank is open top (like a swimming pool) not with a lid on it (like a box)--I just went back and modified the original post.

And, yes, there is enough info to solve the problem. But it's still too hard for people on this list.

Edited August 19, 2010 by Philip Coates

[tjohnson]

And, yes, there is enough info to solve the problem.

Prove it.

[Brant Gaede]

Well, if someone competent in math like GS says "prove it," prove it.

--Brant

Edited August 19, 2010 by Brant Gaede

[DavidMcK]

8 X 8 X 4

[merjet]

Same answer as DavidMcK, 8 x 8 x 4.

[Philip Coates]

Yes! That is correct! YAAY!!

Did you do it by trial and error (graphing, plugging in) or what -method- did you use?

Please show your steps and explain your reasoning.

,,,,

(Meanwhile, I'm going to post the similar but easier problem which I think more people can get as Phil Quiz for Thursday...I think some people will be able to identify and explain the method for that one.

In due course, if no one posts a full explanation I'll post a full explanation for -this- problem. It also includes how all similar problems can be solved.)

Edited August 19, 2010 by Philip Coates

[9thdoctor]

And if the "tank" is not open top? What's the answer then?

Add inability to compose word problems to Phil’s list of incompetencies.

Wait and see if he can admit error again. Robert Campbell suggests that Leonard Peikoff discontinue his podcasts, and I suggest Phil discontinue his Wednesday pop quizzes for the same reasons.

http://www.objectivistliving.com/forums/index.php?showtopic=8901&view=findpost&p=102310

Edited August 19, 2010 by Ninth Doctor

[merjet]

Assume the bottom is square.

height*width*width = 256

surface = width*width + 4*height*width

By substitution:

surface = width*width + 1024/width

Take the derivative, set it to 0, and solve for width. Solution is 8.

height = 256/64 = 4.

Will you now retract your assertion that it's too hard for people on this list (other than Ba'al and Dragonfly)?

Another way to get the answer is use Solver in MS Excel.

Edited August 19, 2010 by Merlin Jetton

[Brant Gaede]

So the top made a difference? If the tank had a top, what would be the answer?

--Brant

Edited August 19, 2010 by Brant Gaede

[9thdoctor]

So the top made a difference? If the tank had a top, what would be the answer?

No top:

8 times 4 times 4 + 8 times 8 = 192

vs.

6.35 times 5 = 202

With top:

8 times 4 times 4 + 8 times 8 times 2 = 256

vs.

6.35 times 6 = 242

Expected result when Phil composes a word problem:

Integral of 1 over (jam) d (jam)

[merjet]

So the top made a difference? If the tank had a top, what would be the answer?

--Brant

No top: Base = 8x8, Height = 4, Surface = 192

With top: Base = 6.35x6.35, Height = 6.35, Surface = 242

[tjohnson]

Assume the bottom is square.

height*width*width = 256

surface = width*width + 4*height*width

By substitution:

surface = width*width + 1024/width

Take the derivative, set it to 0, and solve for width. Solution is 8.

height = 256/64 = 4.

Will you now retract your assertion that it's too hard for people on this list (other than Ba'al and Dragonfly)?

Another way to get the answer is use Solver in MS Excel.

Yes, Merlin also made an assumption, which I think is necessary to use differential calculus to minimize the surface function. Without an assumption you have 3 unknowns and 2 equations.