a wooden cube 9*9*9 has three square holes drilled through it each of which forms a 3*3*9 tunnel through the centre of opposite faces.what is the total surface area of exposed wood

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- Nov 8th 2010, 05:10 AMprasumcube question
a wooden cube 9*9*9 has three square holes drilled through it each of which forms a 3*3*9 tunnel through the centre of opposite faces.what is the total surface area of exposed wood

- Nov 8th 2010, 06:22 AMWilmer
My "brain cells that handle 3D" are weak, few, and far apart.

But here's what I'd do:

1: buy a perfect cube of soft cheese (on sale)

2: turn the 9*9*9 and 3*3*9 to 3*3*3 and 1*1*3

3: "line" the surfaces so that each looks like a "X and O" grid

4: going top to bottom and using appropriate kitchen knife,

remove the vertical 1*3 at center; so three 1*1's have been removed

5: perform similar surgery both ways horizontally; here one must remember

that only two 1*1's are removed for each (the center one is gone from step 4),

so four 1*1's have been removed

a = surface area of initial cube: 6 * 9 = 54

b = "removed area"; a 1*1 from each surface: 6 * 1 = 6

c = "added area"; that's 4 surfaces of six 1*1's: 6 * 4 = 24

So exposed area = a - b + c = 72

Now translate that to a 9*9*9.

Well, hope that helps a bit... - Nov 8th 2010, 11:38 AMearboth
1. Draw a sketch.

2. The surface of the original cube: $\displaystyle a_s = 6 \cdot 9^2 = 486$

3. At each face is missing a square: $\displaystyle 6 \cdot 3^2=54$

4. At each face you find 4 squares which form the walls of the holes: $\displaystyle 6 \cdot 4 \cdot 3^2 = 216$

5. Therefore the complete surface area is: $\displaystyle 486-54+216 = 648$ - Nov 8th 2010, 01:52 PMArchie Meade
Or...

the outside surface area is the area of 8 squares on all 6 faces.

the internal surface area is 4 squares on each of the 6 walls.

$\displaystyle SA=8(6)3^2+4(6)3^2=(48+24)9=(72)9$