The total surface area of a cuboid is 194 m². If its length is 8 m and breadth is 6 m, then what is its volume (in m²) ?

Question

Accepted Answer

To determine the volume of the cuboid, we need to first find its height using the given total surface area, length, and breadth. The formula for the total surface area (TSA) of a cuboid is:

$$ \text{TSA} = 2(lb + bh + hl) $$

where $ l $ is the length, $ b $ is the breadth, and $ h $ is the height.

Given:
- Total surface area (TSA) = 194 m²
- Length ($ l $) = 8 m
- Breadth ($ b $) = 6 m

Substitute these values into the formula:

$$ 194 = 2(8 \times 6 + 6 \times h + h \times 8) $$

Simplify the equation:

$$ 194 = 2(48 + 6h + 8h) $$
$$ 194 = 2(48 + 14h) $$
$$ 194 = 96 + 28h $$

Subtract 96 from both sides:

$$ 98 = 28h $$

Divide both sides by 28 to solve for $ h $:

$$ h = \frac{98}{28} = 3.5 $$

Now that we have the height, we can calculate the volume of the cuboid using the formula:

$$ \text{Volume} = l \times b \times h $$

Substitute the values:

$$ \text{Volume} = 8 \times 6 \times 3.5 $$
$$ \text{Volume} = 48 \times 3.5 $$
$$ \text{Volume} = 168 $$

Thus, the volume of the cuboid is 168 m³, which corresponds to Option 3.

**Explanation of why other options are incorrect:**

1. **Option 1 (112):** This volume would imply a different height calculation, which does not satisfy the given total surface area of 194 m² with the specified length and breadth.

2. **Option 2 (126):** Similar to Option 1, this volume would result from an incorrect height that does not match the given total surface area.

3. **Option 4 (224):** This volume is too large and would require a height that exceeds the calculated height of 3.5 m, thus not aligning with the given total surface area.

Therefore, Option 3 is the correct answer, as it accurately reflects the calculated volume based on the provided dimensions and total surface area.