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Question
A small terrace at a football field comprises 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 (See figure). Calculate the total volume of concrete required to build the terrace.
[Hint: Volume of concrete required to build the first step = `1/4 xx 1/2 xx 50 m^3`]
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Solution
From the figure, it can be observed that
1st step is `1/2` m wide,
2nd step is 1 m wide,
3rd step is `3/2` m wide.
Therefore, the width of each step is increasing by `1/2` m each time whereas their height `1/4` m and length
50 m remains the same.
Therefore, the widths of these steps are
`1/2,1, 3/2, 2`,...
Volume of concrete in 1st step = `1/4 xx1/4 xx50 = 25/4`
Volume of concrete in 2nd step = `1/4 xx 1xx 50 = 50/4`
Volume of concrete in 3rd step = `1/4 xx 3/2 xx 50 = 75/4`
It can be observed that the volumes of concrete in these steps are in an A.P.
`25/4,50/4, 75/4,...`
a = `25/4`
n = 15
and d = `(50/4 - 25/4)`
d = `25/4`
∵ `S_n = n/2[2a + (n - 1)d]`
`S_15 = 15/2[2(25/4)+(15-1)25/4]`
`=15/2[25/2+((14)25)/4]`
`= 15/2[25/2 + 175/2]`
=`15/2(100)`
= 750
Volume of concrete required to build the terrace is 750 m3.
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