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# Cross section of solids and volumes | AIME I 2012 | Question 8

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

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.

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