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Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on cross section of solids and volumes.

## Cross-section of solids and volumes – AIME I, 2012

Cube ABCDEFGH labeled as shown below has edge length 1 and is cut by a plane passing through vertex D and the midpoints M and N of AB and CG respectively. The plane divides the cube into solids. The volume of the larger of the two solids can be written in the form $\frac{p}{q}$ where p and q are relatively prime. find p+q.

• is 107
• is 89
• is 840
• cannot be determined from the given information

### Key Concepts

Calculus

Algebra

Geometry

But try the problem first…

Source

AIME, 2012, Question 8

Calculus Vol 1 and 2 by Apostle

## Try with Hints

First hint

DMN plane cuts the section of solid with $z=\frac{y}{2}-\frac{x}{4}$ intersects base at $y=\frac{x}{2}$

Second Hint

$V=\int_0^1\int_{\frac{x}{2}}^1\int_0^{\frac{y}{2}-\frac{x}{4}}{d}x{d}y{d}z$=$\frac{7}{48}$

Final Step

other portion 1-$\frac{7}{48}$=$\frac{41}{48}$ then 41+48=89.