Question

The volume of concrete required to build the terrace, if
I. the terrace consists of 20 steps each 100 units long.
II. each step has a rise of $\frac{1}{2}m$ and a breadth of $\frac{1}{2}m$.

Answer 1

Given, length of each step is 100 m and width is $\frac{1}{2}m$.
Also, height of 1st step $=\frac{1}{2}m$
Height of 2nd step $\frac{1}{2}+\frac{1}{2}=2\times \frac{1}{2}m$
Height of 3rd step $=\frac{3}{2}m$
and so on.
Let $V_1,V_2,V_3,\cdots V_{20}$, denotes respectively the volume of the concrete required to build the 1st, 2nd, 3rd, ..., 20th step.
Then,
$V_1=(100\times \frac{1}{2}\times \frac{1}{2})m^3{V}_2=[100\times \frac{1}{2}\times (2\times \frac{1}{2})]m^3{V}_3=[100\times \frac{1}{2}\times (3\times \frac{1}{2})]m^3$
and so on.
$\therefore$ Total volume
$=V_1+V_2+\cdots +V_{20}=(100\times \frac{1}{2}\times \frac{1}{2})+[100\times \frac{1}{2}\times (2\times \frac{1}{2})]+[100\times \frac{1}{2}\times (3\times \frac{1}{2})]+\cdots +\left[(100\times \frac{1}{2}\times (20\times \frac{1}{2}))\right]=100\times \frac{1}{2}[\frac{1}{2}+\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+\cdots +\frac{20}{2}]=100\times \frac{1}{2}\times \frac{1}{2}[1+2+\cdots +20]=25\left[\frac{20}{2}\right(2+19\times 1\left)\right]=25\times 10\times 21=5250m^3$
$\therefore$ Both statements are required.