A solid cube has a volume of $1000 \mathrm{~cm}^{3}$. A cube with volume $64 \mathrm{~cm}^{3}$ is removed from one corner of the cube. The resulting solid has a total of nine faces: three large squares that were faces of the original cube, three of the original faces with a square removed from one corner, and three smaller squares. One of the smaller square faces is shaded. The ratio of the area of the shaded face to the surface area of the new solid is of the form $2: x$. What is the value of $x$ ?

![](https://cdn.mathpix.com/cropped/2024_04_17_b6707dc71c8b7ccd30afg-2.jpg?height=477&width=1214&top_left_y=417&top_left_x=518)
The side length of a cube with volume $a^{3} \mathrm{~cm}^{3}$ is $a \mathrm{~cm}$.

Since $10^{3}=1000$ and $4^{3}=64$, the side length of the original cube is $10 \mathrm{~cm}$ and the side length of the smaller cube is $4 \mathrm{~cm}$.

The new figure has nine faces: three $10 \mathrm{~cm} \times 10 \mathrm{~cm}$ squares, three $4 \mathrm{~cm} \times 4 \mathrm{~cm}$ squares, and three faces obtained by removing a $4 \mathrm{~cm} \times 4 \mathrm{~cm}$ square from the corner of a $10 \mathrm{~cm} \times 10 \mathrm{~cm}$ square. Thus, the surface area of the new figure in $\mathrm{cm}^{2}$ is

$$
3\left(10^{2}\right)+3\left(4^{2}\right)+3\left(10^{2}-4^{2}\right)=600
$$

[Note that this is exactly the surface area of the original cube. Can you see a simple reason why this is true?]

The shaded region is a $4 \mathrm{~cm} \times 4 \mathrm{~cm}$ square, so the ratio we seek is $4^{2}: 600$ or $2: 75$.

Therefore, $x=75$.

ANSWER: 75