The internal dimensions of a tank are 12 dm, 8 dm and 5 dm. How many cubes each of edge 7 cm can be placed in the tank with faces prallel to the sides of the tank . Find also, how much space is left unoccupied ?

Options

A. a. `34, 113 dm^(3) `
B. b. `1313, 31.13 dm^(3) `
C. c. `1309, 31.013 dm^(3) `
D. d. `1309 , 13.31 dm^(3) `

Answer

To solve the problem step by step, we will calculate the volume of the tank, determine how many cubes can fit inside it, and find the unoccupied space.

### Step 1: Convert the dimensions of the tank from decimeters to centimeters.
The internal dimensions of the tank are given as:
- Length = 12 dm = 12 × 10 = 120 cm
- Width = 8 dm = 8 × 10 = 80 cm
- Height = 5 dm = 5 × 10 = 50 cm

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

### Step 3: Determine the volume of one cube.
The edge length 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 value:
\[ V_c = 7 \, \text{cm} \times 7 \, \text{cm} \times 7 \, \text{cm} \]
\[ V_c = 343 \, \text{cm}^3 \]

### Step 4: Calculate how many cubes can fit in the tank.
To find out how many cubes can fit in the tank, we need to determine how many cubes can fit along each dimension of the tank:
- Along the length: \( \frac{120 \, \text{cm}}{7 \, \text{cm}} \approx 17.14 \) (we take 17 cubes)
- Along the width: \( \frac{80 \, \text{cm}}{7 \, \text{cm}} \approx 11.43 \) (we take 11 cubes)
- Along the height: \( \frac{50 \, \text{cm}}{7 \, \text{cm}} \approx 7.14 \) (we take 7 cubes)

Now, multiply the number of cubes that can fit along each dimension:
\[ \text{Total cubes} = 17 \times 11 \times 7 = 1309 \]

### Step 5: Calculate the total volume occupied by the cubes.
The total volume occupied by the cubes can be calculated as:
\[ \text{Total volume occupied} = \text{Total cubes} \times V_c \]
Substituting the values:
\[ \text{Total volume occupied} = 1309 \times 343 \]
\[ \text{Total volume occupied} = 448987 \, \text{cm}^3 \]

### Step 6: Calculate the unoccupied space in the tank.
The unoccupied space can be found by subtracting the total volume occupied by the cubes from the volume of the tank:
\[ \text{Unoccupied space} = V - \text{Total volume occupied} \]
Substituting the values:
\[ \text{Unoccupied space} = 480000 \, \text{cm}^3 - 448987 \, \text{cm}^3 \]
\[ \text{Unoccupied space} = 31013 \, \text{cm}^3 \]

### Final Answer:
- Number of cubes that can be placed in the tank: **1309**
- Unoccupied space: **31013 cm³**