Super-Hard SAT Geometry Question 14 | Free SAT Prep

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Question 14·25 Super-Hard SAT Geometry and Trigonometry Questions·Geometry and Trigonometry

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A solid is constructed from $27$ congruent cubes arranged in $3$ layers, each containing $3$ rows and $3$ columns of cubes. The three cubes in the vertical stack at the center of the solid are removed, creating a square tunnel from the top of the solid to the bottom. The volume of the remaining solid is $3000$ cubic units.

All exposed surfaces of the remaining solid, including the interior surfaces of the tunnel, are painted. What is the total painted surface area, in square units?

Your answer

For a solid made of equal cubes, separate the problem into two parts: determine the area of one small face from volume information, then count all exposed faces. When cubes are removed, do not count only the outside faces that disappear; also count any newly exposed interior faces, such as the walls of a tunnel.

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Count the remaining cubes

Start by subtracting the $3$ removed cubes from the original $27$ cubes. Use the remaining total and the given volume to find the volume of one small cube.

Use the cube volume

The cube root of the volume of one small cube gives its edge length. Then square that edge length to find the area of one face.

Account for the tunnel

A $3\times3\times3$ block has $54$ exposed small faces. Consider both the faces lost at the top and bottom openings and the new wall faces exposed inside the tunnel.

Desmos Guide

Calculate the small-cube edge length

Algebraic counting is fastest. For a calculator verification, enter (3000/24)^(1/3) to find the side length of one remaining small cube.

Verify the total area

Enter 64*(5^2) after counting the $64$ exposed small faces: $54$ original outer faces, minus $2$ removed opening faces, plus $12$ interior tunnel-wall faces.

Step-by-step Explanation

Find the volume of one small cube

Originally there are $27$ cubes, but $3$ center cubes are removed. Thus, the remaining solid contains $24$ cubes.

Each small cube has volume

\frac{3000}{24}=125.

Find the side length of each cube

If $s$ is the side length of a small cube, then $s^3=125$ . Therefore, $s=5$ , so each exposed face has area $5^2=25$ square units.

Count exposed faces

Before the tunnel is made, the $3\times3\times3$ solid has $6(3\cdot3)=54$ exposed small faces.

Removing the top and bottom center cubes removes $2$ faces that were originally exposed. The tunnel creates $4$ interior wall faces in each of its $3$ layers, adding $12$ exposed faces. Thus, the remaining solid has

54-2+12=64

exposed small faces.

Calculate the painted area

Multiply the number of exposed faces by the area of each face:

64\cdot25=1600.

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Step-by-step Explanation

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