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 with a hole, volume tells you how much material remains, while surface area counts the faces paint can reach. Use the remaining volume to find one cube’s edge length with a Desmos regression. Then count the outer faces, subtract the openings, and add the tunnel walls. Removing material can expose new surfaces, not only take surfaces away.

Hints

Hint 1
Each small cube has volume equal to its edge length cubed. Because three cubes are removed, $24$ identical cubes make up the given volume. What equation links their shared edge length to $3000$ ?
Hint 2
For surface area, picture the original solid as a large cube with six outside faces, each made of a $3$ -by- $3$ grid. Which squares disappear from the top and bottom when the stack is removed?
Hint 3
The openings aren’t the whole change. Looking down the tunnel, you see four new inner walls. Each runs past three layers of cubes. How many small square faces do those walls add?

Step-by-step

Count the painted small-cube faces
Step 1
Write an equation for the remaining volume
Let $x_1$ be the edge length of one small cube. Removing the three center cubes leaves $27-3=24$ cubes. Each has volume $x_1^3$ , so the given remaining volume means $24x_1^3=3000$ .
Step 2
Find one cube’s edge length
Type $24x_1^3\sim3000$ in Desmos. The $\sim$ tells Desmos to find the value that makes both sides match. Under PARAMETERS, it shows $x_1=5$ , so each small cube has edge length $5$ units.
Step 3
Count the outside squares that remain
Before the removal, the large cube has $6$ outside faces, each a $3$ -by- $3$ grid of small squares. The center stack removes one square from the top and one from the bottom; the other outside squares stay. Type $6(3^2)-2$ in Desmos. It shows $52$ outside squares.
Step 4
Add the newly exposed tunnel walls
The hole has four inner walls, one facing each direction. Each wall exposes one small square face in each of the $3$ layers. Removing cubes exposes these new faces as well as erasing the two openings. Type $52+4(3)$ in Desmos. It shows $64$ painted small square faces altogether.
Step 5
Find the painted area
Each painted square is one face of a small cube, so its area is $x_1^2$ square units. Type $64x_1^2$ in Desmos; it shows $1600$ . The total painted surface area is $1600$ square units. Grid in 1600.

Lessons that teach this

SAT Geometry and TrigonometryIntermediateCoreFind surface area and volume

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

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