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MATH / A solid right prism ABCDEF has a height of 16 and equilateral triangles bases with side length 12, as shown. ABCDEF is sliced with a straight cut through points M, N, P, and Q on e…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

A solid right prism ABCDEFABCDEF has a height of 1616 and equilateral triangles bases with side length 12,12, as shown. ABCDEFABCDEF is sliced with a straight cut through points M,M, N,N, P,P, and QQ on edges DE,DE, DF,DF, CB,CB, and CA,CA, respectively. If DM=4,DM=4, DN=2,DN=2, and CQ=8,CQ=8, determine the volume of the solid QPCDMN.QPCDMN. [asy] pair A, B, C, D, E, F, M,N,P,Q; A=(0,0); B=(12,0); C=(6,-6); D=(6,-22); E=(0,-16); F=(12,-16); M=(2D+E)/3; N=(5D+F)/6; P=(2C+B)/3; Q=(2A+C)/3; draw(A--B--C--A--E--D--F--B--C--D); draw(M--N--P--Q--M, dashed); label("AA", A, NW); label("BB", B, NE); label("CC", C, dir(90)); label("DD", D, S); label("EE", E, SW); label("FF", F, SE); label("MM", M, SW); label("NN", N, SE); label("PP", P, SE); label("QQ", Q, W); label("12", (A+B)/2, dir(90)); label("16", (B+F)/2, dir(0)); [/asy]
Plain-text mathematical notation (without MathML)
A solid right prism ABCDEF has a height of 16 and equilateral triangles bases with side length 12, as shown. ABCDEF is sliced with a straight cut through points M, N, P, and Q on edges DE, DF, CB, and CA, respectively. If DM=4, DN=2, and CQ=8, determine the volume of the solid QPCDMN. [asy]

pair A, B, C, D, E, F, M,N,P,Q;
A=(0,0);
B=(12,0);
C=(6,-6);
D=(6,-22);
E=(0,-16);
F=(12,-16);
M=(2D+E)/3;
N=(5D+F)/6;
P=(2C+B)/3;
Q=(2A+C)/3;

draw(A--B--C--A--E--D--F--B--C--D);
draw(M--N--P--Q--M, dashed);
label("A", A, NW);
label("B", B, NE);
label("C", C, dir(90));
label("D", D, S);
label("E", E, SW);
label("F", F, SE);
label("M", M, SW);
label("N", N, SE);
label("P", P, SE);
label("Q", Q, W);
label("12", (A+B)/2, dir(90));
label("16", (B+F)/2, dir(0));
[/asy]
Original LaTeX notation
A solid right prism $ABCDEF$ has a height of $16$ and equilateral triangles bases with side length $12,$ as shown. $ABCDEF$ is sliced with a straight cut through points $M,$ $N,$ $P,$ and $Q$ on edges $DE,$ $DF,$ $CB,$ and $CA,$ respectively. If $DM=4,$ $DN=2,$ and $CQ=8,$ determine the volume of the solid $QPCDMN.$ [asy]

pair A, B, C, D, E, F, M,N,P,Q;
A=(0,0);
B=(12,0);
C=(6,-6);
D=(6,-22);
E=(0,-16);
F=(12,-16);
M=(2D+E)/3;
N=(5D+F)/6;
P=(2C+B)/3;
Q=(2A+C)/3;

draw(A--B--C--A--E--D--F--B--C--D);
draw(M--N--P--Q--M, dashed);
label("$A$", A, NW);
label("$B$", B, NE);
label("$C$", C, dir(90));
label("$D$", D, S);
label("$E$", E, SW);
label("$F$", F, SE);
label("$M$", M, SW);
label("$N$", N, SE);
label("$P$", P, SE);
label("$Q$", Q, W);
label("12", (A+B)/2, dir(90));
label("16", (B+F)/2, dir(0));
[/asy]

level

Level 5

type

Geometry

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Source and history

task_id
afa76f70-6e29-5d3f-9faf-25c543d4751f
task_key
geometry--test--afa76f70-6e29-5d3f-9faf-25c543d4751f
task_revision_id
1
config / split
geometry / test

Official source

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