Question

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

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Answer 1

Here, we can clearly observe that each step is a cuboid. The length is the same for each

step and is equal to '$50\mathrm{m}$',

The breadth will also be the same for each step and is equal to$\frac{1}{2}\mathrm{m}$, whereas the height for each step is increasing by '$\frac{1}{4}\mathrm{m}$', i.e, for the first step it is$\frac{1}{4}\mathrm{m}$, for the second step it is equal to$\frac{1}{2}\mathrm{m}$, for the third step is $\frac{3}{4}\mathrm{m}$ and so on.

So, to find the total volume of concrete required, we need to find the volume of each step and add it.

Finding the Volume of Steps:

$\begin{array}{rcl}& \Rightarrow & \mathrm{Volume}\mathrm{of}\mathrm{first}\mathrm{step}=\mathrm{Length}\times \mathrm{Breadth}\times \mathrm{Height}\\ & =& 50\times \frac{1}{2}\times \frac{1}{4}\\ & =& 6.25{\mathrm{m}}^3\end{array}$

$\Rightarrow \mathrm{Volume}\mathrm{of}\mathrm{second}\mathrm{step}=50\times \frac{1}{2}\times \frac{1}{2}$ ($\therefore$Height is increasing for each step by $\frac{1}{4}\mathrm{m}$)

$=12.5{\mathrm{m}}^3$

$\begin{array}{rcl}& \Rightarrow & \mathrm{Volume}\mathrm{of}\mathrm{third}\mathrm{step}=50\times \frac{1}{2}\times \frac{3}{4}\\ & =& 18.75{\mathrm{m}}^3\end{array}$

Writing volumes:

$\Rightarrow$ $6.25{\mathrm{m}}^3,12.5{\mathrm{m}}^3,18.75{\mathrm{m}}^3,...$

Here, we can clearly observe that the volume of steps is forming an Arithmetic Progression(A.P.), with the first term($\mathrm{a}$)$=6.25$ and common difference($\mathrm{d}$)$=6.25$

So, finding the sum by applying the formula for the sum of the first $\mathrm{n}$ terms of an A.P.,

$\therefore$ ${\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\left[2\mathrm{a}+\left(\mathrm{n}-1\right)\mathrm{d}\right]$

${\mathrm{S}}_{15}=\frac{15}{2}\left[2\times 6.25+\left(15-1\right)6.25\right]$ ($\because$Here, $\mathrm{n}=15$ because we have to find the volume of $15$ steps)

$$\begin{gathered} =\frac{15}{2}\left[12.5+14\times 6.25\right] \\ =750{\mathrm{m}}^3 \end{gathered}$$

Therefore, the volume of concrete required is $750{\mathrm{m}}^3$