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Home » 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.]
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.]
Arithmetic Progression | CBSE | Class 10 | Maths | NCERT
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.]

Solution:

The first step is \frac{1}{2} m wide, the second step is 1 m wide, and the third step is \frac{3}{2}m wide, as seen in the diagram. When the height reaches \frac{1}{4} m, we can see that the breadth of the step increases by \frac{1}{2}m each time. Furthermore, the length of the steps is always 50m. As a result, the width of steps produces an AP series in the following way:

½ , 1, 3/2, 2, ……..

Volume of steps = Volume of Cuboid

= Length × Breadth Height

Now,

The volume of concrete needed to construct the first step = ¼ ×1/2 ×50 = 25/4

The volume of concrete needed to construct the second step =¼ ×1/×50 = 25/2

The volume of concrete needed to construct the second step = ¼ ×3/2 ×50 = 75/2

Now, we can see the volumes of concrete needed to construct the steps, are in AP series;

25/4 , 25/2 , 75/2 …..

Thus, when we apply the AP series concept,

The first term, a = 25/4

The common difference, d = 25/2 – 25/4 = 25/4

As we all know that, 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 the above equation, we get,

Sn = 15/2 (100)

Sn750

As a result, the total volume of concrete needed to construct the terrace is 750 m3.

i

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