A rectangular tank has length = 4 m, width = 3 m and capacity = `30 m^3`. A small model of the tank is made with capacity `240 cm^3`. Find :
the dimensions of the model. 

Answer

To find the dimensions of the small model of the rectangular tank, we will follow these steps:

### Step 1: Find the height of the original tank.
We know the volume of the rectangular tank is given by the formula:
\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \]
Given:
- Length \( L = 4 \, \text{m} \)
- Width \( W = 3 \, \text{m} \)
- Volume \( V = 30 \, \text{m}^3 \)

We can rearrange the formula to find the height \( H \):
\[ H = \frac{V}{L \times W} = \frac{30}{4 \times 3} = \frac{30}{12} = 2.5 \, \text{m} \]

### Step 2: Determine the scale factor for the model.
The volume of the model is given as \( 240 \, \text{cm}^3 \). First, we need to convert the dimensions of the original tank from meters to centimeters since the model's volume is in cubic centimeters.
- Length \( L = 4 \, \text{m} = 400 \, \text{cm} \)
- Width \( W = 3 \, \text{m} = 300 \, \text{cm} \)
- Height \( H = 2.5 \, \text{m} = 250 \, \text{cm} \)

Let the scale factor be \( k \). The dimensions of the model will be:
- Length of the model = \( 400k \)
- Width of the model = \( 300k \)
- Height of the model = \( 250k \)

### Step 3: Set up the equation for the volume of the model.
The volume of the model can also be expressed as:
\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \]
Substituting the dimensions of the model:
\[ 240 = (400k) \times (300k) \times (250k) \]
This simplifies to:
\[ 240 = 400 \times 300 \times 250 \times k^3 \]

### Step 4: Calculate \( k^3 \).
Calculating the constant:
\[ 400 \times 300 = 120000 \]
\[ 120000 \times 250 = 30000000 \]
Now we have:
\[ 240 = 30000000 \times k^3 \]
Rearranging gives:
\[ k^3 = \frac{240}{30000000} = \frac{240}{30000000} = \frac{1}{125000} = 0.00000192 \]
To simplify:
\[ k^3 = \frac{1}{125} \Rightarrow k = \sqrt[3]{\frac{1}{125}} = \frac{1}{5} \]

### Step 5: Find the dimensions of the model.
Now we can find the dimensions of the model:
- Length of the model = \( 400k = 400 \times \frac{1}{5} = 80 \, \text{cm} \)
- Width of the model = \( 300k = 300 \times \frac{1}{5} = 60 \, \text{cm} \)
- Height of the model = \( 250k = 250 \times \frac{1}{5} = 50 \, \text{cm} \)

### Final Answer:
The dimensions of the model are:
\[ \text{Length} = 80 \, \text{cm}, \quad \text{Width} = 60 \, \text{cm}, \quad \text{Height} = 50 \, \text{cm} \]

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