Question 5

The total volume of concrete is the sum of volumes of individual concrete blocks, which form an arithmetic progression.

Step 1 — Calculate volumes of the first few steps.

We find the volume of concrete for the first step. Then we find the volume for the second step. We will see a pattern.

The length of each step is 50 m. The depth (tread) of each step is $\frac{1}{2}\text{ m}$. The rise (height) of each step is $\frac{1}{4}\text{ m}$.

Volume of the first step ($V_1$): $V_1 = 50 \times \frac{1}{2} \times \frac{1}{4}$

$= 25 \times \frac{1}{4}$

$= \frac{25}{4}\text{ m}^3$

Volume of the second step ($V_2$): $V_2 = 50 \times \frac{1}{2} \times \left(2 \times \frac{1}{4}\right)$

$= 25 \times \frac{2}{4}$

$= \frac{50}{4}\text{ m}^3$

Volume of the third step ($V_3$): $V_3 = 50 \times \frac{1}{2} \times \left(3 \times \frac{1}{4}\right)$

$= 25 \times \frac{3}{4}$

$\boxed{\frac{75}{4}\text{ m}^3}$

Step 2 — Identify the arithmetic progression.

The volumes of the steps form an arithmetic progression. We identify the first term and the common difference.

The sequence of volumes is $\frac{25}{4}, \frac{50}{4}, \frac{75}{4}, \dots$. The first term $a = \frac{25}{4}\text{ m}^3$. The common difference $d = \frac{50}{4} - \frac{25}{4}$.

$d = \frac{25}{4}\text{ m}^3$

There are 15 steps in total.

Step 3 — Calculate the total volume.

We use the formula for the sum of an arithmetic progression. The sum of $n$ terms is $S_n = \frac{n}{2}[2a + (n-1)d]$.

Here, $n = 15$, $a = \frac{25}{4}$, and $d = \frac{25}{4}$.

$S_{15} = \frac{15}{2} \left[2 \left(\frac{25}{4}\right) + (15-1) \left(\frac{25}{4}\right)\right]$

$S_{15} = \frac{15}{2} \left[\frac{50}{4} + 14 \left(\frac{25}{4}\right)\right]$

$S_{15} = \frac{15}{2} \left[\frac{50}{4} + \frac{350}{4}\right]$

$S_{15} = \frac{15}{2} \left[\frac{400}{4}\right]$

$S_{15} = \frac{15}{2} \times 100$

$S_{15} = 15 \times 50$

$\boxed{750\text{ m}^3}$

Answer

The total volume of concrete required to build the terrace is 750 m³.