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
Observation from the figure:
The widths of the steps are as follows:
1st step: \( \frac 12 \) m
2nd step: 1 m
3rd step: \( \frac 32 \) m
The width of each step increases by \( \frac 12 \) m consecutively.
The height of each step is \( \frac 14 \) m, and the length is 50 m, both remaining constant.
The sequence of step widths is: \( \frac 12, 1, \frac 32, 2, \dotsb \)
The volume of concrete for each step is calculated as:
1st step: \( \frac 14 \times \frac 12 \times 50 = \frac {25}{4} \) m\(^3\)
2nd step: \( \frac 14 \times 1 \times 50 = \frac {25}{2} \) m\(^3\)
3rd step: \( \frac 14 \times \frac 32 \times 50 = \frac {75}{4} \) m\(^3\)
The volumes of concrete for these steps form an arithmetic progression (A.P.).
The A.P. of volumes is: \( \frac {25}{4}, \frac {25}{2}, \frac {75}{4}, \dotsb \)
The first term of the A.P. is \( a = \frac {25}{4} \).
The common difference is \( d = \frac {25}{2} - \frac {25}{4} = \frac {25}{4} \).
The formula for the sum of an A.P. is \( S_n = \frac n2[2a + (n-1)d] \).
To find the total volume for 15 steps (\( S_{15} \)):
\( S_{15} = \frac {15}{2}[2(\frac {25}{4}) + (15-1)\frac {25}{4}] \)
