Question

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> A small terrace at a football ground comprises $15$ steps each of which is $50m$ long and built of solid concrete.
Each step has a rise of $4/1m$ and a tread of $2/1m.$ Calculate the total volume of concrete required to build the terrace.

[Hint : Volume of concrete required to bulid the first step $=\frac{1}{4}\times \frac{1}{2}\times 50m^3$]

Answer 1

As we can see from the given figure, the first step is is $1/2$ m wide, $2^{nd}$ step is $1m$ wide and $3^{rd}$ step is $3/2$m wide.
Thus we can understand that the width of step by $1/2m$ each time when height is $1/4m$. And also, given length of the steps is $50m$ all the time. So, the width of steps forms a series $AP$ in such a way that; $1/2,1,3/2,2,..........$

Volume of steps = Volume of Cuboid
$=\text{Length}\times \text{Breadth Height}$
Now,
Volume of concrete required to build the first step $=\frac{1}{4}\times 1/2\times 50=25/4$
Volume of concrete required to build the second step $=\frac{1}{4}\times 1/\times 50=25/2$
Volume of concrete required to build the second step $=\frac{1}{4}\times 3/2/\times 50=75/2$
Now, we can see the volumes of concrete required to build the steps, are in $AP$ series;
$25/4,25/2,75/2......$
Thus, applying the $AP$ series concept,
First term, $a=25/4$
Common difference, $d=25/2-25/4=25/4$
As we know, the sum of $n$ terms is ;
$S_n=n/2[2a+(n-1\left)d\right]=15/2(2\times 25/4)+(15/2-1)25/4)$
Upon solving, we get
$S_n=15/2\left(100\right)$
$S_{n}750$
Hence, the total volume of concrete required to build the terrace is $750m^3.$