NCERT Solutions
Class 10 - Mathematics - Chapter 5: ARITHMETIC PROGRESSIONS - Exercise 5.4 (Optional)

Question 5

Step 1: Understand the shape of each step.

Each step is like a solid rectangular block with:

• Length (along the terrace) = 50 m
• Rise (height) = \(\dfrac{1}{4}\) m
• Tread (depth) = \(\dfrac{1}{2}\) m

But as we go up, each new step extends one more tread further back than the one below it.

Step 2: Write the volume of each step.

For a rectangular block:

\[\text{Volume} = \text{length} \times \text{breadth} \times \text{height}\]

First step:

Depth = \(\dfrac{1}{2}\) m, height = \(\dfrac{1}{4}\) m, length = 50 m.

\[V_1 = 50 \times \dfrac{1}{2} \times \dfrac{1}{4} = 50 \times \dfrac{1}{8} = \dfrac{50}{8}\,\text{m}^3\]

Second step:

It occupies two treads in depth: \(2 \times \dfrac{1}{2} = 1\) m.

\[V_2 = 50 \times 1 \times \dfrac{1}{4} = 50 \times \dfrac{1}{4} = \dfrac{50}{4}\,\text{m}^3\]

Third step:

Depth = \(3 \times \dfrac{1}{2} = \dfrac{3}{2}\) m.

\[V_3 = 50 \times \dfrac{3}{2} \times \dfrac{1}{4} = 50 \times \dfrac{3}{8} = \dfrac{150}{8}\,\text{m}^3\]

Continuing in this way, the \(n\)th step has depth \(\dfrac{n}{2}\) m, so:

\[V_n = 50 \times \dfrac{n}{2} \times \dfrac{1}{4} = 50 \times \dfrac{n}{8} = \dfrac{50}{8}n = \dfrac{25}{4}n\,\text{m}^3\]

Step 3: Sum the volumes of all 15 steps.

We need:

\[V_{\text{total}} = V_1 + V_2 + \cdots + V_{15}\]

Using the formula for \(V_n\):

\[V_{\text{total}} = \sum_{n=1}^{15} \dfrac{25}{4}n = \dfrac{25}{4} \sum_{n=1}^{15} n\]

Now,

\[\sum_{n=1}^{15} n = \dfrac{15 \cdot 16}{2} = 120\]

So,

\[V_{\text{total}} = \dfrac{25}{4} \times 120 = 25 \times 30 = 750\,\text{m}^3\]

Conclusion

The total volume of concrete required to build the terrace is \(750\,\text{m}^3\).