The length, breadth and height of a cboid are in the ratio 1:2:3. If they are increased by 100%, 200% and 200% respectively, then compared to the original volume the increase in the volume of the cuboid will be

Options

A. 5 times
B. 18 times
C. 12 times
D. 17 times

Answer

To solve the problem, we need to follow these steps:

### Step 1: Define the dimensions of the cuboid
Let the length (L), breadth (B), and height (H) of the cuboid be represented in terms of a variable based on the given ratio of 1:2:3. We can express them as:
- Length (L) = x
- Breadth (B) = 2x
- Height (H) = 3x

### Step 2: Calculate the original volume of the cuboid
The volume (V) of a cuboid is given by the formula:
\[ V = L \times B \times H \]
Substituting the values we have:
\[ V = x \times 2x \times 3x = 6x^3 \]

### Step 3: Determine the new dimensions after the increase
The dimensions are increased by:
- Length: Increased by 100% → New Length = \( x + 100\% \text{ of } x = x + x = 2x \)
- Breadth: Increased by 200% → New Breadth = \( 2x + 200\% \text{ of } 2x = 2x + 4x = 6x \)
- Height: Increased by 200% → New Height = \( 3x + 200\% \text{ of } 3x = 3x + 6x = 9x \)

### Step 4: Calculate the new volume of the cuboid
Now we can calculate the new volume (V') using the new dimensions:
\[ V' = (2x) \times (6x) \times (9x) \]
\[ V' = 108x^3 \]

### Step 5: Calculate the increase in volume
To find the increase in volume, we subtract the original volume from the new volume:
\[ \text{Increase in Volume} = V' - V = 108x^3 - 6x^3 = 102x^3 \]

### Step 6: Calculate the percentage increase in volume
To find the percentage increase in volume compared to the original volume:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase in Volume}}{\text{Original Volume}} \right) \times 100 \]
\[ \text{Percentage Increase} = \left( \frac{102x^3}{6x^3} \right) \times 100 = 17 \times 100 = 1700\% \]

### Final Answer
The increase in the volume of the cuboid compared to the original volume is **1700%**.

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