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 m3]
![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 m3]](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/5-4-q5-1565182253.png)
Solution:
The sum of the first n terms of an AP is given by Sₙ = 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 1st 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) is the same for each step that is ½ m
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 m3
Volume of concrete in 3rd step = 50m × ½ m × ¾ m = 18.75 m3
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 the 15th value since number of steps = 15
Sum of n terms, Sₙ = n/2 [2a + (n - 1) d]
S₁₅ = 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.
☛ Check: NCERT Solutions Class 10 Maths Chapter 5
Video Solution:
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 m³]
Class 10 Maths NCERT Solutions Chapter 5 Exercise 5.4 Question 5
Summary:
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.
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