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 $\dfrac{1}{4}$ m and a tread of $\dfrac{1}{2}$ m. Calculate the total volume of concrete required to build the terrace.

From the figure, it can be observed that the height of 1st step is $\dfrac{1}{4}$ m

Height of the 2nd step is $\Big(\dfrac{1}{4} + \dfrac{1}{4}\Big) m = \dfrac{1}{2} m$

Height of 3rd step is $\Big(\dfrac{1}{4} + \dfrac{1}{4} + \dfrac{1}{4}\Big) m = \dfrac{3}{4} m$

Therefore, the height of each step is increasing by $\dfrac{1}{4}$ m

Length = 50 m and width (tread) is the same for each step that is $\dfrac{1}{2}$ m

Volume of step can be considered as Volume of Cuboid = length × breadth × height

Volume of concrete in 1st step = 50 m × $\dfrac{1}{2}$ m × $\dfrac{1}{4}$ m = 6.25 m³

Volume of concrete in 2nd step = 50m × $\dfrac{1}{2}$ m × $\dfrac{1}{2}$ m = 12.50 m³

Volume of concrete in 3rd step = 50m × $\dfrac{1}{2}$ m × $\dfrac{3}{4}$ m = 18.75 m³

Volumes : 6.25 m³, 12.50 m³, 18.75 m³, ….

The above list is an A.P.

First term (a) = 6.25, Common difference (d) = 12.50 - 6.25 = 6.25

By formula,

Sum of n terms = S_n = $\dfrac{n}{2}[2a + (n - 1)d]$

Substituting values we get :

$S_{15} = \dfrac{15}{2}[2 \times 6.25 + (15 - 1) \times 6.25] \\[1em] = \dfrac{15}{2}[12.50 + 14 \times 6.25] \\[1em] = \dfrac{15}{2}[12.50 + 87.50] \\[1em] = \dfrac{15}{2} \times 100 \\[1em] = 15 \times 50 \\[1em] = 750.$

Hence, total volume of concrete required to build terrace = 750 m³.