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
1^st step is \(\frac 12\) m wide,
2^nd step is 1 m wide,
3^rd 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 \(750\  m^3\).