The internal length, breadth and height of a box are 30cm, 24cm and 15cm. Find the largest number of cubes which can be placed inside this box if the edge of each cube is
(i) 3cm (ii) 4cm (iii) 5cm

Answer

To solve the problem, we need to find the largest number of cubes that can fit inside a box with given dimensions. The dimensions of the box are:

- Length = 30 cm
- Breadth = 24 cm
- Height = 15 cm

We will calculate the number of cubes that can fit for each given edge length of the cubes (3 cm, 4 cm, and 5 cm).

### Step-by-step Solution:

1. **Calculate the Volume of the Box:**
\[ \text{Volume of the box} = \text{Length} \times \text{Breadth} \times \text{Height} = 30 \, \text{cm} \times 24 \, \text{cm} \times 15 \, \text{cm} \]
\[ = 10800 \, \text{cm}^3 \]

2. **(i) For cubes with edge length 3 cm:**
- Calculate the volume of one cube:
\[ \text{Volume of one cube} = \text{Edge}^3 = 3 \, \text{cm} \times 3 \, \text{cm} \times 3 \, \text{cm} = 27 \, \text{cm}^3 \]
- Calculate the number of cubes that can fit in the box:
\[ \text{Number of cubes} = \frac{\text{Volume of the box}}{\text{Volume of one cube}} = \frac{10800 \, \text{cm}^3}{27 \, \text{cm}^3} = 400 \]

3. **(ii) For cubes with edge length 4 cm:**
- Calculate the volume of one cube:
\[ \text{Volume of one cube} = 4 \, \text{cm} \times 4 \, \text{cm} \times 4 \, \text{cm} = 64 \, \text{cm}^3 \]
- Calculate the number of cubes that can fit in the box:
\[ \text{Number of cubes} = \frac{\text{Volume of the box}}{\text{Volume of one cube}} = \frac{10800 \, \text{cm}^3}{64 \, \text{cm}^3} = 168.75 \]
Since we cannot have a fraction of a cube, we take the whole number:
\[ \text{Number of cubes} = 168 \]

4. **(iii) For cubes with edge length 5 cm:**
- Calculate the volume of one cube:
\[ \text{Volume of one cube} = 5 \, \text{cm} \times 5 \, \text{cm} \times 5 \, \text{cm} = 125 \, \text{cm}^3 \]
- Calculate the number of cubes that can fit in the box:
\[ \text{Number of cubes} = \frac{\text{Volume of the box}}{\text{Volume of one cube}} = \frac{10800 \, \text{cm}^3}{125 \, \text{cm}^3} = 86.4 \]
Again, since we cannot have a fraction of a cube, we take the whole number:
\[ \text{Number of cubes} = 86 \]

### Final Answers:
- (i) For edge length 3 cm: **400 cubes**
- (ii) For edge length 4 cm: **168 cubes**
- (iii) For edge length 5 cm: **86 cubes**