Question

A small terrace at a hockey ground comprises of 10 steps each of which is 20m long and built of solid concrete.each step has a rise of 1/44 and a trend of 1/2m.calculate the total volume of concrete required to build the terrace

Answer 1

||✪✪ CORRECT QUESTION ✪✪||

A small terrace at a hockey ground comprises of 10 steps each of which is 20m long and built of solid concrete.each step has a rise of 1/4 and a trend of 1/2m.calculate the total volume of concrete required to build the terrace ?

|| ✰✰ ANSWER ✰✰ ||

❁❁ Refer To Image First .. ❁❁

In image , we can see that ,, length of First step is 20m (Always), and height is 1/4m and breadth is 1/2m (always).

so, we can say that :-

➺ Volume of 1st step = L * B * H = 20 * (1/4) * (1/2) = (5/2)m³

Now, Lets head to the 2nd step,,

we can see that, here also, length is 20m , and height is this time (1/4 + 1/4)m and breadth is 1/2m.

So,

➻ Volume of 2nd step :- 20 * (1/4 + 1/4) * 1/2 = 20 * 1/2 * 1/2 = 5 m³ .

Now, lets Check 3rd step also,,

From diagram we can say that, again length is same as 20m, breadth is also same as 1/2 m , and this time our height is (1/4 + 1/4 + 1/4)m.

So,

➼ Volume of 3rd Step :- 20 * 1/2 * (1/4 + 1/4 + 1/4) = 20 * 1/2 * 3/4 = (15/2) m³ .

_________________________

with this we can say that now, volume of Each step as (5/2 m³) , (5 m³) , (15/2 m³) ___________ upto 10 steps are in AP.

where,

➾ First term = a = (5/2)

➾ common difference = a2 - a1 = (5) - (5/2) = (5/2)

➾ n = 10 steps .

So,

Sₙ = (n/2) [ 2a + (n-1)d ]

putting all values we get Now,,,

☛ S₁₀ = (10/2) [ 2 *(5/2) + (10-1)(5/2) ]

☛ S₁₀ = 5 [ 5 + 9*5/2 ]

☛ S₁₀ = 5 [ 5 + (45/2) ]

☛ S₁₀ = 5 [ (55/2 ]

☛ S₁₀ = (275/2)

☛ S₁₀ = 137.5 m³

Hence, we can say that, the total volume of concrete required to build the terrace is 137.5m³.

Answer 2

$\huge{\boxed{\boxed{Answer:}}}$

Given:

• A small terrace at a hockey ground comprises of 10 steps each of which is 20 m long and built of solid concrete.

• Each step has a rise of 1/4 and a trend of 1/2 m.

Find:

• Find the total volume of concrete required to build the terrace.

Calculations:

$\sf = 10 (1/2) = number \: of \: steps. \\ </p><p>\sf = 20 \: m \: (1/4) = length \: of \: steps. \\ </p><p>\sf = 1/4 = increase \: of \: steps. \\ </p><p>\sf = 1/2 = tread \: (step) \: of \: others.$

Formula:

Total volume of concrete needed = sum of volume of each step.

Volume of cuboid:

$\sf = Volume = Base \: area × Height. \\ </p><p>\sf= Length × Base × Height$

Finding the volume of the first step:

$\sf = 20 \times \frac{1}{4} \times \frac{1}{2} \: m {}^{3}$

$\sf = 2.5 \: m {}^{2}$

$\sf = Height = 2 \times Height \\ \sf = Height = 2 \times \frac{1}{2} \: m = 1 \: m$

Now, let's find the second step, using the same volume formula:

$\sf = Height = 20 \times \frac{1}{4} \times 1 \: m {}^{3} \\ \sf = \frac{20}{4} \\ \sf = 5 \: m {}^{3}$

Finding the third step, using the same volume formula:

$\sf = 3 \times \frac{1}{2} = \frac{3}{2} \: m \\ \sf = 20 \times \frac{1}{4 } \times \frac{3}{2} \: m {}^{3} \\ \sf \sf = 7.5$

The volumes are 1, 5, 7.5.........etc.

$\sf=(i) \: st \: term=1 \\ \sf=(ii) \: (1) \: and \: (2) \: term = 1 - 5 = 4\\ \sf=(iii) \: (2) \: and \: (3) \: term = 5 - 7.5 =2.5$

Consecutive terms are not equal:

Finding for next 10 steps:

$\sf=Sn = (n/2) [2a + (n-1) d]$

Adding values to the above equation:

$\sf=S10 = \frac{10}{2}\: \frac{2×5}{2}+(10-1)\frac{5}{2}$

$\sf=S10 = S\frac{ (S + 9 × 5)}{2}$

$\sf=S10 = S [(5 + \frac{45}{2)}]$

$\sf=S10 = 5\frac{55}{2}$

$\sf=S10 = \frac{275}{2}$

$\sf=S10 = 137.5 m^2$

Therefore, 137.5 m² is the total volume of concrete required to build the terrace.