![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 1 4 m and a tread of 1 2 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 ×1/2 ×50 m3.] Ncert solutions class 10 chapter 5-17 Q.5](https://ask.truemaths.com/wp-content/uploads/2021/07/A-small-terrace-at-a-football-ground-comprises-of-15-steps-each-of-which-is-50-m-long-and-built-of-solid.png)
The best question of optional chapter of arithmetic progressions of exercise 5.4 of class 10th math, how i solve this question in easy way 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 1 4 m and a tread of 1 2 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 ×1/2 ×50 m3.] Ncert solutions class 10 chapter 5-17
As we can see from the given figure, the first step is ½ m wide, 2nd step is 1m wide and 3rd step is 3/2m wide. Thus we can understand that the width of step by ½ m each time when height is ¼ m. And also, given length of the steps is 50m all the time. So, the width of steps forms a series AP in such a way that;
½ , 1, 3/2, 2, ……..
Volume of steps = Volume of Cuboid
= Length × Breadth Height
Now,
Volume of concrete required to build the first step = ¼ ×1/2 ×50 = 25/4
Volume of concrete required to build the second step =¼ ×1/×50 = 25/2
Volume of concrete required to build the second step = ¼ ×3/2 ×50 = 75/2
Now, we can see the volumes of concrete required to build the steps, are in AP series;
25/4 , 25/2 , 75/2 …..
Thus, applying the AP series concept,
First term, a = 25/4
Common difference, d = 25/2 – 25/4 = 25/4
As we know, the sum of n terms is;
Sn = n/2[2a+(n-1)d] = 15/2(2×(25/4 )+(15/2 -1)25/4)
Upon solving, we get,
Sn = 15/2 (100)
Sn750
Hence, the total volume of concrete required to build the terrace is 750 m3.