# Q : 5     A small terrace at a football ground comprises of  $\small 15$ steps each of which is $\small 50$ m long and   built  of solid concrete. Each step has a rise of   $\small \frac{1}{4}\: m$   and a tread of   $\small \frac{1}{2}\: m$ . (see Fig. $\small 5.8$).  Calculate the total volume of concrete required to build the terrace.                                 [Hint : Volume of concrete required to build the first step  $\small =\frac{1}{4}\times \frac{1}{2}\times 50\: m^3$  ]

G Gautam harsolia

It is given that
football ground comprises of  $\small 15$ steps each of which is $\small 50$ m long and Each step has a rise of   $\small \frac{1}{4}\: m$   and a tread of   $\small \frac{1}{2}\: m$
Now,
Volume required to make first step = $\frac{1}{4}\times \frac{1}{2}\times 50 = 6.25 \ m^3$

Similarly,

Volume required to make 2nd step = $\left ( \frac{1}{4}+\frac{1}{4}\right )\times \frac{1}{2}\times 50=\frac{1}{2}\times \frac{1}{2}\times 50 = 12.5 \ m^3$
And
Volume required to make 3rd step = $\left ( \frac{1}{4}+\frac{1}{4}+\frac{1}{4}\right )\times \frac{1}{2}\times 50=\frac{3}{4}\times \frac{1}{2}\times 50 = 18.75 \ m^3$

And so on
We can clearly see that this is an AP with $a= 6.25 \ and \ d = 6.25$
Now,  total volume of concrete required to build the terrace of 15 such step is
$S_{15} =\frac{15}{2}\left \{ 2 \times 6.25 +(15-1)6.25 \right \}$
$S_{15} =\frac{15}{2}\left \{ 12.5 +87.5\right \}$
$S_{15} =\frac{15}{2}\times 100$
$S_{15} =15\times 50 = 750$
Therefore,  total volume of concrete required to build the terrace of 15 such step is $750 \ m^3$

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