The internal dimensions of a box are 1.2 m. 80 cm and 50 cm. How many cubes each of edge 7 cm can be packed in the box with faces parallel to the sides of the box. Also, find the space left empty in the box.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Convert all dimensions to the same unit
The internal dimensions of the box are given as:
- Length = 1.2 m = 120 cm (since 1 m = 100 cm)
- Width = 80 cm
- Height = 50 cm

### Step 2: Calculate the volume of the box
The volume \( V \) of the box can be calculated using the formula:
\[ V = \text{Length} \times \text{Width} \times \text{Height} \]
Substituting the values:
\[ V = 120 \, \text{cm} \times 80 \, \text{cm} \times 50 \, \text{cm} \]
Calculating this gives:
\[ V = 480000 \, \text{cm}^3 \]

### Step 3: Calculate the volume of one cube
The edge of each cube is given as 7 cm. The volume \( V_c \) of one cube can be calculated using the formula:
\[ V_c = \text{edge}^3 \]
Substituting the edge length:
\[ V_c = 7 \, \text{cm} \times 7 \, \text{cm} \times 7 \, \text{cm} = 343 \, \text{cm}^3 \]

### Step 4: Determine how many cubes can fit in the box
To find out how many cubes can fit, we need to calculate how many cubes can fit along each dimension of the box.

- Along the length (120 cm):
\[ \text{Number of cubes along length} = \frac{120}{7} \approx 17.14 \implies 17 \, \text{cubes} \]

- Along the width (80 cm):
\[ \text{Number of cubes along width} = \frac{80}{7} \approx 11.43 \implies 11 \, \text{cubes} \]

- Along the height (50 cm):
\[ \text{Number of cubes along height} = \frac{50}{7} \approx 7.14 \implies 7 \, \text{cubes} \]

### Step 5: Calculate the total number of cubes
The total number of cubes that can fit in the box is the product of the number of cubes along each dimension:
\[ \text{Total number of cubes} = 17 \times 11 \times 7 = 1309 \, \text{cubes} \]

### Step 6: Calculate the total volume occupied by the cubes
The total volume occupied by the cubes can be calculated as:
\[ \text{Total volume of cubes} = \text{Number of cubes} \times \text{Volume of one cube} \]
Substituting the values:
\[ \text{Total volume of cubes} = 1309 \times 343 = 448907 \, \text{cm}^3 \]

### Step 7: Calculate the empty space left in the box
The empty space left in the box can be calculated by subtracting the total volume of the cubes from the volume of the box:
\[ \text{Empty space} = \text{Volume of box} - \text{Total volume of cubes} \]
Substituting the values:
\[ \text{Empty space} = 480000 \, \text{cm}^3 - 448907 \, \text{cm}^3 = 31193 \, \text{cm}^3 \]

### Final Answers
- The number of cubes that can be packed in the box is **1309**.
- The space left empty in the box is **31193 cm³**.