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Construction Dilemma

Math brain teasers require computations to solve.

 Puzzle ID: #21741 Fun: (1.96) Difficulty: (2.96) Category: Math Submitted By: Sane Corrected By: MarcM1098

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?

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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