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 $\frac 14$ m and a tread of $\frac 12$ m. (see Fig. 5.8). Calculate the total volume of concrete required to build the terrace.Fig. 5.8

Answer 1

From the figure, it can be observed that
1st step is $\frac 12$ m wide,
2nd step is 1 m wide,
3rd step is $\frac 32$ m wide.
Therefore, the width of each step is increasing by $\frac 12$ m each time Whereas their height $\frac 14$ m and length 50 m remains the same.
Therefore, the widths of these steps are
$\frac 12, 1, \frac 32, 2, ......$
Volume of concrete in 1st step $= \frac 14 \times \frac 12 \times 50 = \frac {25}{4}$

Volume of concrete in 2nd step $= \frac 14 \times 1 \times 50 = \frac {25}{2}$

Volume of concrete in 3rd step $= \frac 14 \times \frac 32 \times 50 = \frac {75}{4}$
It can be observed that the volumes of concrete in these steps are in an A.P.

$\frac {25}{4}, \frac {25}{2}, \frac {75}{4}, ......$

$a = \frac {25}{4}$

$d = \frac {25}{2} - \frac {25}{4} = \frac {25}{4}$

And, $S_n = \frac n2[2a + (n-1)d]$

$S_{15} = \frac {15}{2}[2(\frac {25}{4}) + (15-1)\frac {25}{4}]$

$S_{15} = \frac {15}{2}[\frac {25}{2} + 14\times \frac {25}{4}]$

$S_{15} = \frac {15}{2}[\frac {25}{2} + \frac {175}{2}]$

$S_{15} = \frac {15}{2} \times 100$
$S_{15} = 750$

So, Volume of concrete required to build the terrace is $750\ m^3$.