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.

Cross section of solids and volumes
  • is 107
  • is 89
  • is 840
  • cannot be determined from the given information

Key Concepts


Calculus

Algebra

Geometry

Check the Answer


But try the problem first…

Answer: 89.

Source
Suggested Reading

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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