Length of a room is 1$\frac{1}{2}$ times its breadth. If its height be 3.3 m and volume be 123$\frac{3}{4}$ m³, find the length of the room :

Question

Accepted Answer

To determine the length of the room, we need to use the given information about the room's dimensions and volume. Let's break down the problem step by step:

1. **Understanding the Relationship Between Length and Breadth:**
   - The length of the room is given as $1\frac{1}{2}$ times its breadth. This can be expressed as:
     $$
     \text{Length} = \frac{3}{2} \times \text{Breadth}
     $$

2. **Using the Volume Formula:**
   - The volume of a rectangular room is calculated using the formula:
     $$
     \text{Volume} = \text{Length} \times \text{Breadth} \times \text{Height}
     $$
   - We are given the volume as $123\frac{3}{4} \, \text{m}^3$, which is equivalent to $\frac{495}{4} \, \text{m}^3$.
   - The height of the room is given as $3.3 \, \text{m}$.

3. **Setting Up the Equation:**
   - Substitute the known values into the volume formula:
     $$
     \frac{3}{2} \times \text{Breadth} \times \text{Breadth} \times 3.3 = \frac{495}{4}
     $$
   - Simplify the equation:
     $$
     \frac{3}{2} \times \text{Breadth}^2 \times 3.3 = \frac{495}{4}
     $$
   - Multiply both sides by 2 to eliminate the fraction:
     $$
     3 \times \text{Breadth}^2 \times 3.3 = \frac{495}{2}
     $$
   - Divide both sides by 3.3:
     $$
     3 \times \text{Breadth}^2 = \frac{495}{2 \times 3.3}
     $$
   - Calculate the right side:
     $$
     3 \times \text{Breadth}^2 = \frac{495}{6.6} = 75
     $$
   - Divide both sides by 3:
     $$
     \text{Breadth}^2 = 25
     $$
   - Take the square root of both sides:
     $$
     \text{Breadth} = 5 \, \text{m}
     $$

4. **Calculating the Length:**
   - Since the length is $1\frac{1}{2}$ times the breadth:
     $$
     \text{Length} = \frac{3}{2} \times 5 = 7.5 \, \text{m}
     $$

5. **Conclusion:**
   - The calculated length of the room is $7.5 \, \text{m}$, which matches Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (7.2 m):** This option is incorrect because, based on the calculations, the length derived from the given volume and dimensions does not match 7.2 m.
- **Option 3 (8.2 m):** This option is incorrect as the calculated length is 7.5 m, not 8.2 m.
- **Option 4 (8.4 m):** This option is incorrect because the correct length, as calculated, is 7.5 m, not 8.4 m.

Thus, the correct answer is indeed Option 1, which is 7.5 m.