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.) . Calculate the total volume of concrete required to build the terrace. [Hint : Volume of concrete required to build the first step = 1/4 ×1/2 ×50 m3.]

Ncert solutions class 10 chapter 5-17

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 know that the width of step by ½ m each time when height is ¼ m. And the 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, ……..

The volume of steps = Volume of Cuboid

= Length × Breadth Height

Now,

The volume of concrete required to build the first step

= ¼ ×1/2 ×50 = 25/4

The volume of concrete required to build the second step

=¼ ×1/×50 = 25/2

The volume of concrete required to build the second step

= ¼ ×3/2 ×50 = 75/2

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.

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