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The Cut Cube (Posted on 2019-02-13) Difficulty: 3 of 5
A cube on a table has edge length 24. A plane intersects the cube's four vertical edges at points A, B, C, and D such that point A is a vertex of the cube lying on the table. The heights of points B and C from the table floor are 7 and 12, respectively.

Calculate the volume of the portion of the cube that lies underneath the cutting plane.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution solution | Comment 2 of 4 |
Assuming A, B, C and D are either in clockwise or anticlockwise order (B and C on adjacent vertical cells):

The plane is linear in z as a function of x and y (taking the base of the cube as lying on the z = 0 plane).

We have a point at (0,0,0); another at (24,0,7); another at (34,34,12). The fourth is at (0,24,5), with the 5 having been calculated as the same distance above zero as 12 is above 7.

Taking one face of the block whose volume is to be found as the right triangle with legs of length 24 and 5, and the opposite face as the trapezoid with edge 7 opposite that with length 12, the cross-sectional area, increases linearly from 60 at the former face to 228 at the latter face.

The average cross section is therefore 144 square units. Multiplied by the 24-unit length we get 3456 as the volume of the block.

  Posted by Charlie on 2019-02-13 12:25:24
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