5. A wooden rectangular prism has edges of length 8, 10, and 15. From two of its vertices, one cube each is cut out, with the lengths of their edges being natural numbers. First, a cube of the largest possible volume is cut from the prism, and then a cube of the smallest possible volume. What percentage of the prism's volume is cut out?

## Problems worth 10 points:
5. A wooden rectangular prism has edges of length 8, 10, and 15. From two of its vertices, one cube each is cut out, with the lengths of the edges being natural numbers. First, a cube of the largest possible volume is cut out from the prism, and then a cube of the smallest possible volume. What percentage of the volume of the prism has been cut out?

## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_81794be492b317694280g-20.jpg?height=702&width=557&top_left_y=871&top_left_x=247)

Since the lengths of the edges of the cut-out cubes are natural numbers, the cube of the largest possible volume has an edge length of 8 cm, and the cube of the smallest possible volume has an edge length of 1 cm.

1 POINT

One possible way is shown in the figure on the right.

The volume of the prism is $8 \cdot 10 \cdot 15 = 1200 \, \text{cm}^3$.

1 POINT

The volume of the cube with the largest possible volume is $8^3 = 512 \, \text{cm}^3$. 1 POINT

The volume of the cube with the smallest possible volume is $1^3 = 1 \, \text{cm}^3$.

1 POINT

The volume of the cut-out part is $513 \, \text{cm}^3$.

The percentage of the cut-out part is $\frac{513}{1200} = \frac{171}{400} = 42.75\%$. 2 POINTS

![](https://cdn.mathpix.com/cropped/2024_05_30_81794be492b317694280g-20.jpg?height=686&width=557&top_left_y=1573&top_left_x=1246)

TOTAL 6 POINTS

Note: If a student incorrectly concludes that a "boundary part of the prism" must remain and uses 7 instead of 8 for the edge length of the cube with the largest possible volume, 1 POINT is deducted, and the rest is graded according to the "follows the error" principle.

However, for other incorrect values of the edge lengths of the smallest or largest possible cubes, 2 or 3 POINTS are deducted (as these are larger errors in understanding the problem), 1 POINT for determining these lengths and 1 POINT for each incorrectly determined volume. The rest is graded according to the "follows the error" principle.