The edges of a cuboid are in the ratio 1:2:3 and its surface area is `88 cm^(2)`. The volume of the cuboid is :

Options

A. `120 cm^(3)`
B. `64 cm^(3)`
C. `48 cm^(3)`
D. `24 cm^(3)`

Answer

To find the volume of the cuboid given that its edges are in the ratio 1:2:3 and its surface area is 88 cm², we can follow these steps:

### Step 1: Define the dimensions of the cuboid
Let the dimensions of the cuboid be:
- Length (l) = x
- Breadth (b) = 2x
- Height (h) = 3x

### Step 2: Write the formula for the surface area of the cuboid
The surface area (SA) of a cuboid is given by the formula:
\[ SA = 2(lb + bh + lh) \]

### Step 3: Substitute the dimensions into the surface area formula
Substituting the values of l, b, and h into the surface area formula:
\[ SA = 2(x \cdot 2x + 2x \cdot 3x + 3x \cdot x) \]
\[ = 2(2x^2 + 6x^2 + 3x^2) \]
\[ = 2(11x^2) \]
\[ = 22x^2 \]

### Step 4: Set the surface area equal to 88 cm²
Now, we set the surface area equal to the given value:
\[ 22x^2 = 88 \]

### Step 5: Solve for x²
Dividing both sides by 22:
\[ x^2 = \frac{88}{22} \]
\[ x^2 = 4 \]

### Step 6: Solve for x
Taking the square root of both sides:
\[ x = 2 \]

### Step 7: Find the dimensions of the cuboid
Now we can find the dimensions:
- Length (l) = x = 2 cm
- Breadth (b) = 2x = 4 cm
- Height (h) = 3x = 6 cm

### Step 8: Calculate the volume of the cuboid
The volume (V) of the cuboid is given by the formula:
\[ V = l \cdot b \cdot h \]
Substituting the dimensions:
\[ V = 2 \cdot 4 \cdot 6 \]
\[ = 48 \text{ cm}^3 \]

### Final Answer
The volume of the cuboid is **48 cm³**.

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