A small terrace at a uphill temple has 100 steps each of which is 108 m long and built of solid concrete. Each step has a rise of 1/3 m and has a tread of 2/3 m. The total volume of the concrete required to build the terrace will be

Options

A. 188100 `m^(3)`
B. 256000 `m^(3)`
C. 144000 `m^(3)`
D. 121200 `m^(3)`

Answer

To find the total volume of concrete required to build the terrace with 100 steps, we can break down the problem into a series of steps. Each step has specific dimensions, and the volume of each step can be calculated using the formula for the volume of a cuboid.

### Step-by-Step Solution:

1. **Identify the dimensions of each step:**
- Length (L) = 108 m
- Rise (Height, H) = 1/3 m for the first step
- Tread (Width, B) = 2/3 m

2. **Calculate the volume of the first step:**
- Volume (V) = Length × Breadth × Height
- V₁ = 108 m × (2/3) m × (1/3) m
- V₁ = 108 × (2/3) × (1/3) = 108 × 2 / 9 = 24 m³

3. **Determine the dimensions for subsequent steps:**
- The height of each subsequent step increases by 1/3 m.
- Therefore, the height for the second step (H₂) = 2/3 m
- The height for the third step (H₃) = 1 m (3/3 m)

4. **Calculate the volume of the second step:**
- V₂ = 108 m × (2/3) m × (2/3) m
- V₂ = 108 × (2/3) × (2/3) = 108 × 4 / 9 = 48 m³

5. **Calculate the volume of the third step:**
- V₃ = 108 m × (2/3) m × (1) m
- V₃ = 108 × (2/3) × 1 = 108 × 2 / 3 = 72 m³

6. **Identify the pattern in the volumes:**
- The volumes of the steps form an arithmetic progression (AP):
- V₁ = 24 m³
- V₂ = 48 m³
- V₃ = 72 m³
- The common difference (d) = 24 m³.

7. **Find the total volume for all 100 steps:**
- The formula for the sum of the first n terms of an AP is:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
- Here, \( n = 100 \), \( a = 24 \), and \( d = 24 \).
- Substitute the values:
\[ S_{100} = \frac{100}{2} \times (2 \times 24 + (100 - 1) \times 24) \]
\[ S_{100} = 50 \times (48 + 99 \times 24) \]
\[ S_{100} = 50 \times (48 + 2376) = 50 \times 2424 = 121200 m³ \]

### Final Answer:
The total volume of concrete required to build the terrace is **121200 m³**.