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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 1/4 m and a tread of 1/2 m. Calculate the total volume of concrete required to build the terrace

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  ]

                

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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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