Exam question I couldn't get right

[A Tripolation]

In my logics class today, we had our midterm. On it was a bonus that I couldn't figure out. It reminded me of something from Calc III (multivariable calculus), but I couldn't nail it down. The question went something like this:

 

Let B be a box, made from spatial dimensions L, length, W, width, and H, height. Let S be all the points around the box that are, at most, 1 unit away from the box. Express S in terms of L, W, and H.

 

How should I have solved this? I had a picture where there was a cube, with a larger, sphere-like object, because the maximums would curve around the vertices of the cube. Did I start it right?

[DrRocket]

In my logics class today, we had our midterm. On it was a bonus that I couldn't figure out. It reminded me of something from Calc III (multivariable calculus), but I couldn't nail it down. The question went something like this:

 

Let B be a box, made from spatial dimensions L, length, W, width, and H, height. Let S be all the points around the box that are, at most, 1 unit away from the box. Express S in terms of L, W, and H.

 

How should I have solved this? I had a picture where there was a cube, with a larger, sphere-like object, because the maximums would curve around the vertices of the cube. Did I start it right?

 

You have the relative complement of a box of dimensions L+2, W+2, H+2 and a box of dimensions L, W, H (I assume that "around the box" implies "outside of the box) -- a solid rectangle with another solid rectangle removed from its center.

 

This is not so much an exercise in logic as an exerecise in converting imprecise language into precise language. One has to figure out what the speaker actually means, which can be subject to interpretation.

 

 

Edit: The comments below are correct. The outer box is actually a rectanglular solid in which the edges and corners are rounded off, with a radius of curvature of 1.

Edited March 4, 2011 by DrRocket

[imatfaal]

Not sure if DocRoc is right there - I agree with you Trip that the corners would have to be curved (the outside corner is sqrt3 away from inside corner). I would say that you have a collection of shapes

2 @ W*H*1

2 @ W*D*1

2 @ H*W*1

 

4 @ H*1*1

4 @ W*1*1

4 @ D*1*1*

 

8 @ unit demi-demi-demi-spheres - ie an eight corner of a sphere -

 

you could then add these groups together together to get

 

1 w*h*2

1 h*d*2

1 w*d*2

 

1 h*1*4

1 w*1*4

1 d*1*4

 

1 unit sphere

 

I don't think you can simplify any more

 

Matthew

[Sisyphus]

Not sure if DocRoc is right there - I agree with you Trip that the corners would have to be curved (the outside corner is sqrt3 away from inside corner). I would say that you have a collection of shapes

2 @ W*H*1

2 @ W*D*1

2 @ H*W*1

 

4 @ H*1*1

4 @ W*1*1

4 @ D*1*1*

 

8 @ unit demi-demi-demi-spheres - ie an eight corner of a sphere -

 

you could then add these groups together together to get

 

1 w*h*2

1 h*d*2

1 w*d*2

 

1 h*1*4

1 w*1*4

1 d*1*4

 

1 unit sphere

 

I don't think you can simplify any more

 

Matthew

 

Don't forget that the "edge" pieces would also be curved, having a quarter circle cross section. Since there are 4 of each kind, you can just do it as four cylinders.

 

The answer as I see it would be:

 

2wh + 2wd + 2hd + wπ + hπ + dπ + 4π/3

[Mr Skeptic]

First of all, it matters somewhat if the box is considered solid, or whether your roundish object might be hollow. It also matters how you want to answer the problem, which you could do as a formula or using set notation. And if you're using sets, can you use infinitely many simpler sets or would you want to limit yourself to fewer but slightly more complicated sets?

[imatfaal]

Don't forget that the "edge" pieces would also be curved, having a quarter circle cross section. Since there are 4 of each kind, you can just do it as four cylinders.

 

The answer as I see it would be:

 

2wh + 2wd + 2hd + wπ + hπ + dπ + 4π/3

 

Of Course! Damn - I missed that.

 

Mr Skeptik - " Let S be all the points around the box that are, at most, 1 unit away from the box." I think the question precludes just a surface - a point half a unit away from the original box is definitely within the question.

Edited March 4, 2011 by imatfaal

[Mr Skeptic]

Mr Skeptik - " Let S be all the points around the box that are, at most, 1 unit away from the box." I think the question precludes just a surface - a point half a unit away from the original box is definitely within the question.

 

Right, but if the box is hollow and more than 2 units wide, you'd have a volume that is hollow on the inside. (but my guess would be that the box should be solid)

[A Tripolation]

Thanks for all the replies guys. The box is definitely solid. But its size is arbitrary. We needed an abstract equation that would always be able to determine S if B is known. I take it Sisyphus' simplification of imatfaal's answer is the solution to this problem? It's a bit confusing, but I'll try to reconstruct it on a different problem.