### Brain Teasers

# Construction Dilemma

Mr. Math, or Sane, was hired by Nortel to design the dimensions for their building. He decided to make the building an open cylindrical area inscribed in a perfect cube.

But a tragic incident caused Sane to become sick and not be able to communicate! The only dimension that he had written down was for the corner office, 16 by 8 meters (which extends to the edge of the open area).

The construction workers can not start over now, as they've already begun building. But if they want to finish the Nortel building in time, they will need to know the final dimension of the perfect cube and cylinder! Can anyone here help them?

Alternate: If the construction workers became ambitious and decided to make all the offices out of solid wood, approximately how much would it cost them at $10 per m^3?

But a tragic incident caused Sane to become sick and not be able to communicate! The only dimension that he had written down was for the corner office, 16 by 8 meters (which extends to the edge of the open area).

The construction workers can not start over now, as they've already begun building. But if they want to finish the Nortel building in time, they will need to know the final dimension of the perfect cube and cylinder! Can anyone here help them?

Alternate: If the construction workers became ambitious and decided to make all the offices out of solid wood, approximately how much would it cost them at $10 per m^3?

### Hint

Formulate an equation where the hypotenuse of the two squared unknown distances will equal the radius of the building.### Answer

Formulate an equation where the hypotenuse of the two squared unknown distances will equal the radius of the building.(Since our answer only needs two dimensions, we will convert them into square (building), circle (open area), and rectangle (corner office).

[The Pythagorean theorem will solve for the root of the radius]

Let a represent the length of the rectangle, then x-a be the radius of the circle minus the length of the rectangle, same to b (to width of the rectangle):

a^2 + b^2 = x^2

(x - a)^2 + (x - b)^2 = x^2

x^2 = (x - 16)(x - 16) + (x - 8)(x - 8)

[Use foil to expand the brackets]

x^2 = (x^2 - 16x - 16x + 256) + (x^2 - 8x - 8x + 64)

x^2 = 2x^2 - 48x + 320

[Order as a quadratic equation]

x^2 - 48x + 320 = 0

[Factor the equation using the quadratic formula with X excluded]

Let a equal 1, b equal 48, and c equal 320 as temporary variables in the factoring equation:

x = -b{+/-}[b^2 - 4ac]^1/2 / 2a

x = -48{+/-}[2304 - 1280]^1/2 / 2

x = -48{+/-}32 / 2

x = -40 and x = -8

[Input those numbers into the now factored equation]

(x - 40)(x - 8) = 0

If the radius were 8, then that would mean the corner office would end up extending into the circular area (because the corner office is 16 x 8).

Therefore, the diameter of the cylinder or the dimensions of the building must be 40 * 2 (radius x 2) to make 80m x 80m.

Alternate: The volume of the building is expressed by:

v = l * w * h

But remember, since the cylinder is open space, we need to figure it too.

The volume of the cylinder is expressed by:

v = pi(r^2) * h

Therefore, the volume of solid wood is the volume of the building less the empty area:

v = [l * w * h] - [pi(r^2) * h]

v = [80 * 80 * 80] - [pi(40^2) * 80]

v ~= 512 000 - 402124

v ~= 109 876 m^3

If wood is $10 per m^3:

10 * 109876 = $1 098 760

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## Comments

uh..way to complicated...my brain doesnt work that well..

If anyone wants me to take out the altenate, just ask.

Figuring out the dimensions is relatively easy, but the alternate question is confusing. Making offices, to me, means building walls, but you can't answer that with the information provided. The alternate question is actually asking for the volume of the office space.

No, it's not making offices. It's "turning all the offices to solid wood". In other words, the total area of the cube, subtract the open area in the middle.

DUH THIS WUZ SOOOOOOOOOO EASY U JUST HAV TO ADD THE SQUARE RADIOUS X PIE = THE LATERAL SIDE OF THE CYLENDER..........FROM THERE U JUST DO SIMPLE MATH ...................................................................................................................................................................................................................................................................o, and ya i dont hav ne clue how to figure it out

This was easy saying that we just went over this in class and its one of the few things that I am good at.

whoa!!!! There is no way i could figure this out.....it's sssssoooooooooooooooooooo hard!!!!!

This is more like a math problem from one of those math competition tests I took in high school. those tests where you get better credit for not even answering the question than for answering it wrong. i would have skipped this question.

I'm not a mathmatition but I know a decent amouint. I also know buildings.

Now for this offic there is nothing to state that it is bisected by the open area just that it extends to it. Now wit that have been stated it would touch partway down on the circle not the top or side so to speak.

with it like this you would have a bigger area to deal with. more like the circle pluss say 8ft past that at least. Now as far as the building is concerned 8ft isnt that much 6inch walls = -1 foot each side so you end up with a room thats 7ft by 15ft long ehough but fairly narrow.Any desk will come halfway accross the room.

structurally speaking it could be built but It wouldn't be able to be used for much.

Now for this offic there is nothing to state that it is bisected by the open area just that it extends to it. Now wit that have been stated it would touch partway down on the circle not the top or side so to speak.

with it like this you would have a bigger area to deal with. more like the circle pluss say 8ft past that at least. Now as far as the building is concerned 8ft isnt that much 6inch walls = -1 foot each side so you end up with a room thats 7ft by 15ft long ehough but fairly narrow.Any desk will come halfway accross the room.

structurally speaking it could be built but It wouldn't be able to be used for much.

I am sorry, and I guess this is why I am not an architect, but I am having trouble visualizing this structure. The way I am conceptualizing it is, at first, 2D, looking at it from above, seeing a square with a circle inscribed in it. in the small space between circle and square corner, i have a rectangle office measuring 16x8 - is that ANYwhere on cue? If not, please explain where I went terrible wrong, how to fix it, and then maybe I can approach the problem correctly and see how "hard" it really is.

I ended up with some quadratic eqn using sines and cosines, which actually made the numbers a bit more manageable somehow. Good fun! I got it!

Not enough detail in question. What is the office extends to open area? In which direction? The author tries to explain in the comments about the area of the cube? I think a blueprint is needed to convey these dimensions, not a vague description.

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