#### Question

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 figure) calculate the total volume of concrete required to build the terrace.

[Hint : Volume of concrete required to build the first step = 1/4 x 1/2 x 50m^{3}]

#### Solution

From the figure, it can be observed that

1^{st} step is 1/2 m wide,

2^{nd} step is 1 m wide,

3^{rd} step is 3/2 m wide.

Therefore, the width of each step is increasing by 1/2 m each time whereas their height 1/4 m and length

50 m remains the same.

Therefore, the widths of these steps are

1/2,1, 3/2, 2,...

Volume of concrete in 1^{st} step = `1/4 xx1/4 xx50 = 25/4`

Volume of concrete in 2^{nd} step = `1/4 xx 1xx 50 = 25/2`

Volume of concrete in 3^{rd} step = `1/4 xx 3/2 xx 50 = 75/4`

It can be observed that the volumes of concrete in these steps are in an A.P.

`25/4,25/2, 75/4,...`

a = 25/4

d = 25/2 - 25/4 = 25/4

and

`S_n = n/2[2a + (n - 1)d]`

`S_15 = 15/2[2(25/4)+(15-1)25/4]`

`=15/2[25/2+((14)25)/4]`

`= 15/2[25/2 + 175/2]`

=15/2(100) = 750

Volume of concrete required to build the terrace is 750 m^{3}.