Question

A staircase consists of 20 steps. The length and breadth of each step is 100 $m$ and $\frac{1}{2}m$ respectively. The height of the lowermost step is $\frac{1}{2}m$ and the height of the subsequent step increases by $\frac{1}{2}m$. Find the volume in $m^3$ of the concrete required to make the staircase.

• A. 5250
• B. 2520
• C. 5225
• D. 5520

Answer 1

The correct option is A

5250

Given, length of each step is 100m and width is $\frac{1}{2}m$.
Also, height of 1st step $=\frac{1}{2}m$
Height of IInd step $\frac{1}{2}+\frac{1}{2}=2\times \frac{1}{2}m$
Height of IIIrd 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
$=V_1+V_2+\cdots +V_{20}=(100\times \frac{1}{2}\times \frac{1}{2})+[100\times \frac{1}{2}\times (2\times \frac{1}{2})]+[100\times \frac{1}{2}\times (3\times \frac{1}{2})]+\cdots +\left[(100\times \frac{1}{2}\times (20\times \frac{1}{2}))\right]=100\times \frac{1}{2}[\frac{1}{2}+\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+\cdots +\frac{20}{2}]=100\times \frac{1}{2}\times \frac{1}{2}[1+2+\cdots +20]=25\left[\frac{20}{2}\right(2+19\times 1\left)\right]=25\times 10\times 21=5250m^3$