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 ratio between the total surface area of the tank and its model.

Answer

To find the ratio between the total surface area of the rectangular tank and its model, we will follow these steps:

### Step 1: Calculate the height of the tank
The volume (capacity) of the tank is given by the formula:
\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \]
Given:
- Length = 4 m
- Width = 3 m
- Volume = 30 m³

We can rearrange the formula to find the height:
\[ \text{Height} = \frac{\text{Volume}}{\text{Length} \times \text{Width}} = \frac{30}{4 \times 3} = \frac{30}{12} = 2.5 \text{ m} \]

### Step 2: Calculate the total surface area of the tank
The total surface area (TSA) of a rectangular prism is given by the formula:
\[ \text{TSA} = 2(\text{Length} \times \text{Width} + \text{Width} \times \text{Height} + \text{Height} \times \text{Length}) \]
Substituting the values:
\[ \text{TSA}_{\text{tank}} = 2(4 \times 3 + 3 \times 2.5 + 2.5 \times 4) \]
Calculating each term:
- \(4 \times 3 = 12\)
- \(3 \times 2.5 = 7.5\)
- \(2.5 \times 4 = 10\)

Now substituting back:
\[ \text{TSA}_{\text{tank}} = 2(12 + 7.5 + 10) = 2(29.5) = 59 \text{ m}^2 \]

### Step 3: Calculate the total surface area of the model
The volume of the model is given as 240 cm³. We will assume dimensions that satisfy this volume. Let's take:
- Length = 15 cm
- Width = 8 cm
- Height = 2 cm

Now, we can verify the volume:
\[ \text{Volume}_{\text{model}} = 15 \times 8 \times 2 = 240 \text{ cm}^3 \]

Now, calculate the total surface area of the model using the same TSA formula:
\[ \text{TSA}_{\text{model}} = 2(15 \times 8 + 8 \times 2 + 2 \times 15) \]
Calculating each term:
- \(15 \times 8 = 120\)
- \(8 \times 2 = 16\)
- \(2 \times 15 = 30\)

Now substituting back:
\[ \text{TSA}_{\text{model}} = 2(120 + 16 + 30) = 2(166) = 332 \text{ cm}^2 \]

### Step 4: Convert the surface area of the tank to cm²
Since 1 m² = 10,000 cm², we convert the surface area of the tank:
\[ \text{TSA}_{\text{tank}} = 59 \text{ m}^2 = 59 \times 10,000 = 590,000 \text{ cm}^2 \]

### Step 5: Calculate the ratio of the total surface areas
Now we can find the ratio of the total surface area of the tank to that of the model:
\[ \text{Ratio} = \frac{\text{TSA}_{\text{tank}}}{\text{TSA}_{\text{model}}} = \frac{590,000 \text{ cm}^2}{332 \text{ cm}^2} \]

Calculating the ratio:
\[ \text{Ratio} \approx 1771:1 \]

### Final Answer
The ratio between the total surface area of the tank and its model is approximately \(1771:1\).

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