MathDB

A cube-shaped container has vertices

A

B

C

, and

D

where

\overline{AB}

and

\overline{CD}

are parallel edges of the cube, and

\overline{AC}

and

\overline{BD}

are diagonals of the faces of the cube. Vertex

A

of the cube is set on a horizontal plane

\mathcal P

so that the plane of the rectangle

ABCD

is perpendicular to

\mathcal P

, vertex

B

2

meters above

\mathcal P

, vertex

C

8

meters above

\mathcal P

, and vertex

D

10

meters above

\mathcal P

. The cube contains water whose surface is

7

meters above

\mathcal P

. The volume of the water is

\tfrac mn

cubic meters, where

m

and

n

are relatively prime positive integers. Find

m+n

. [asy] size(250); defaultpen(linewidth(0.6)); pair A = origin, B = (6,3), X = rotate(40)*B, Y = rotate(70)*X, C = X+Y, Z = X+B, D = B+C, W = B+Y; pair P1 = 0.8*C+0.2*Y, P2 = 2/3*C+1/3*X, P3 = 0.2*D+0.8*Z, P4 = 0.63*D+0.37*W; pair E = (-20,6), F = (-6,-5), G = (18,-2), H = (9,8); filldraw(E--F--G--H--cycle,rgb(0.98,0.98,0.2)); fill(A--Y--P1--P4--P3--Z--B--cycle,rgb(0.35,0.7,0.9)); draw(A--B--Z--X--A--Y--C--X^^C--D--Z); draw(P1--P2--P3--P4--cycle^^D--P4); dot("

A

",A,S); dot("

B

",B,S); dot("

C

",C,N); dot("

D

",D,N); label("

\mathcal P

",(-13,4.5)); [/asy]

Problem Statement