A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of and a tread of . (see Fig. 5.8). Calculate the total volume of concrete required to build the terrace. [Hint : Volume of concrete required to build the first step = ]

Understand the Question The total volume of concrete required is the sum of the volumes of all 15 individual steps. Each step is a cuboidal block with length , tread (width) , and height increasing by for each successive step. The volumes of the steps form an Arithmetic Progression (AP), where the sum of terms is given by . Step 1 · Calculate Volumes of the First Few Steps Given: Volume of the 1st step (): Volume of the 2nd step (): Volume of the 3rd step (): Step 2 · Identify the Arithmetic Progression The sequence of volumes is This forms an AP with: First term, Common difference, Total number of steps, Step 3 · Calculate the Total Volume of Concrete Using the sum formula : Answer > Common Mistakes Height Increment Error: Forgetting that each successive step is built from the ground up, meaning the height of the step is , not a constant . Formula Substitution: Misidentifying or as instead of the step volume .

Arithmetic Progressions | Exercise 5.4

Question 5

A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete.

Each step has a rise of $\dfrac{1}{4} \text{ m}$ and a tread of $\dfrac{1}{2} \text{ m}$ . (see Fig. 5.8). Calculate the total volume of concrete required to build the terrace.

[Hint : Volume of concrete required to build the first step = $\dfrac{1}{4} \times \dfrac{1}{2} \times 50 \text{ m}^3$ ]

Solution

Understand the Question

The total volume of concrete required is the sum of the volumes of all 15 individual steps.
Each step is a cuboidal block with length $= 50\text{ m}$ , tread (width) $= \dfrac{1}{2}\text{ m}$ , and height increasing by $\dfrac{1}{4}\text{ m}$ for each successive step.
The volumes of the steps form an Arithmetic Progression (AP), where the sum of $n$ terms is given by $S_n = \dfrac{n}{2}[2a + (n - 1)d]$ .

Step 1 · Calculate Volumes of the First Few Steps

Given:

$\text{Length} = 50\text{ m}$
$\text{Tread (width)} = \dfrac{1}{2}\text{ m}$
$\text{Rise (height of 1st step)} = \dfrac{1}{4}\text{ m}$

Volume of the 1st step ( $V_1$ ):

\begin{aligned} V_1 &= 50 \times \dfrac{1}{2} \times \dfrac{1}{4} \\[0.6em] &= 25 \times \dfrac{1}{4} \\[0.6em] &= \dfrac{25}{4} \text{ m}^3 \end{aligned}

Volume of the 2nd step ( $V_2$ ):

\begin{aligned} V_2 &= 50 \times \dfrac{1}{2} \times \left(2 \times \dfrac{1}{4}\right) \\[0.6em] &= 25 \times \dfrac{2}{4} \\[0.6em] &= \dfrac{50}{4} \text{ m}^3 \end{aligned}

Volume of the 3rd step ( $V_3$ ):

\begin{aligned} V_3 &= 50 \times \dfrac{1}{2} \times \left(3 \times \dfrac{1}{4}\right) \\[0.6em] &= 25 \times \dfrac{3}{4} \\[0.6em] &= \dfrac{75}{4} \text{ m}^3 \end{aligned}

Step 2 · Identify the Arithmetic Progression

The sequence of volumes is $\dfrac{25}{4}, \dfrac{50}{4}, \dfrac{75}{4}, \dots$

This forms an AP with:

First term, $a = \dfrac{25}{4} \text{ m}^3$
Common difference, $d = \dfrac{50}{4} - \dfrac{25}{4} = \dfrac{25}{4} \text{ m}^3$
Total number of steps, $n = 15$

Step 3 · Calculate the Total Volume of Concrete

Using the sum formula $S_n = \dfrac{n}{2}[2a + (n - 1)d]$ :

\begin{aligned} S_{15} &= \dfrac{15}{2} \left[2 \left(\dfrac{25}{4}\right) + (15 - 1) \left(\dfrac{25}{4}\right)\right] \\[0.8em] &= \dfrac{15}{2} \left[\dfrac{50}{4} + 14 \left(\dfrac{25}{4}\right)\right] \\[0.8em] &= \dfrac{15}{2} \left[\dfrac{50}{4} + \dfrac{350}{4}\right] \\[0.8em] &= \dfrac{15}{2} \left[\dfrac{400}{4}\right] \\[0.8em] &= \dfrac{15}{2} \times 100 \\[0.6em] &= 15 \times 50 \\[0.6em] &= 750 \text{ m}^3 \end{aligned}

Answer

$750\text{ m}^3$

Common Mistakes

Height Increment Error: Forgetting that each successive step is built from the ground up, meaning the height of the $n^{\text{th}}$ step is $n \times \dfrac{1}{4}\text{ m}$ , not a constant $\dfrac{1}{4}\text{ m}$ .
Formula Substitution: Misidentifying $a$ or $d$ as $\dfrac{1}{4}$ instead of the step volume $\dfrac{25}{4}\text{ m}^3$ .

Question 5

More questions in Exercise 5.4