Question

A small terrace at a football field comprises 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 xx 1/2 xx 50  m^3`]

Answer 1

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 = 50/4`

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.

`25/4,50/4, 75/4,...`

a = `25/4`

n = 15

and d = `(50/4 - 25/4)`

d = `25/4`

∵ `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 m³.