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 $\frac{1}{2}\ m$ and a tread of $\frac{1}{2}\ m$. Calculate the total volume of concrete required to build the terrace.

Given:

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 $\frac{1}{2}\ m$ and a tread of $\frac{1}{2}\ m$.

 To do:

We have to calculate the total volume of concrete required to build the terrace.

Solution:

The length of a step $=50\ m$

Breadth of a step $=\frac{1}{2}\ m$

Height of a step $=\frac{1}{4}\ m$

Therefore,

Volume of concrete required to build one step $=(50 \times \frac{1}{4} \times \frac{1}{2})\ m^{3}$

Total volume of concrete required to build the terrace $=50 \times \frac{1}{2}[\frac{1}{4}+\frac{2}{4}+\frac{3}{4}+\ldots+\frac{15}{4}]$

$=25[\frac{1}{4}+\frac{2}{4}+\frac{3}{4}+\ldots+\frac{15}{4}]$

$=\frac{25}{4}[1+2+3+\ldots+15]$

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

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

$=\frac{25}{4} \times \frac{15}{2} \times 16$

$=750\ m^{3}$

Hence, the total volume of concrete required to build the terrace is $750\ m^{3}$.