# 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 ¼ m and a tread of ½ 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 =¼ × ½ × 50 m^{3}]

Fig. 5.8

**Solution:**

The sum of the first n terms of an AP is given by S_{n} = n/2 [2a + (n - 1) d], where a is the first term, d is a common difference and n is the number of terms.

- From the figure, it can be observed that the Height of 1
^{st}step is ¼ m - Height of the 2nd step is (¼ + ¼) m = ½ m
- Height of 3rd step is (¼ + ¼ + ¼) m = 3/4 m

Therefore, the height of each step is increasing by ¼ m

- Length 50m and width (tread) 1 m remain the same for each of the steps:

Volume of Step can be considered as Volume of Cuboid = length × breadth × height

Volume of concrete in 1st step = 50m × ½ m × ¼ m = 6.25 m³

Volume of concrete in 2nd step = 50m × ½ m × ½ m = 12.50 m^{3}

Volume of concrete in 3rd step = 50m × ½ m × ¾ m = 18.75 m^{3}

It can be observed that the volumes of concrete in these steps are in an A.P.: 6.25m³, 12.50m³, 18.75m³, ....

First term a = 6.25, Common difference d = 6.25

Let nth term of AP be number of steps = 15

Sum of n terms, S_{n} = n/2 [2a + (n - 1) d]

S_{15} = 15 / 2 [2 × 6.25 + (15 - 1) × 6.25]

= 15 / 2 [12.50 + 14 × 6.25]

= 15 / 2 [12.50 + 87.50]

= 15/2 × 100

= 750 m³

Therefore, the volume of concrete required to build the terrace is 750 m³.

**Video Solution:**

### Class 10 Maths NCERT Solutions - Chapter 5 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 ¼ m and a tread of ½ m . (see Fig. 5.8). Calculate the total volume of concrete required to build the terrace

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 ¼ m and a tread of ½ m, then the total volume of concrete required to build the terrace is 750 cubic meters.