A small terrace at a flat football ground has 15 steps, each of which is 50 m long and built of solid concrete. Each step has a rise of
4
1
m and a tread of
2
1
m.Calculate the total volume of concrete required to built the terrace.
Note: You cannot build a step in midair. You have to fill the air beneath it.
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Though it was very easy, I favourited this question only because it is related to football!!
true you said Vaibhav setia, i had derived a formula for it earlier, but this time i did it in a bit different way much easier then my own past soln.
question taken from class 10 ncert chapter INDIA chapter name - arithmetic progression exercise - optional
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Nice work finding it!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1
Are you studying in 10th?
practically we don't built that way.
Yess I did it right!!
I used same method too. :)
NOW I CAN UNDERSTAND...
I don't get it at first..X_X now I understand.. tricky question :D
ok
nice ullu bnaawing,,,,,,,,, ans is 750
brilliant
perfect simple solution..... very nice
Think out of box method..superb
this is brilliant depiction! totally out of the box, actually 'inside the box' ! kudos
This method is do easy.....!
At first step we need to construct 1 block, at second step 2 blocks and third step 3 blocks and so on up to 15th step.
Total no of Blocks = 1+2+3+.......+15 = 15*(15+1)/2 = 120
Volume of one block = 50 1/2 1/4 = 6.25 m^3
Total volume = 120 x 6.25 = 750 m^3
WRIGHT....
this is also what i did.
First, you have to solve for how many rectangular sections there are. There will be 1 for the first step, 2 for the second, all the way until the 15th step. Thus there are a total of 120 blocks of 5 0 × 0 . 5 × 0 . 2 5 dimensions. Therefore the area is 7 5 0 .
The volume of the Concrete steps can be viewed as a prism.
Therefore its volume can be calculated by the following formula:
Volume = Cross-sectional Area * Depth
The Cross-sectional area can be calculated with the help of triangular numbers, As the number is 15 the following equation is pretty straight forward:
2 n ( n + 1 ) → 2 1 5 × 1 6 = 1 2 0 .
This formula calculates for a unit, whose value is one, so evidently it needs to be scaled down. In this case the unit is an 8 1 of the square metre.
C r o s s s e c t i o n a l a r e a = 1 2 0 × 8 1 = 1 5 m 2
Finally plugging the 50 metres into the formula gives the answer: V o l u m e = 5 0 × 1 5 = 7 5 0 m 3
{(15 16)/2} {50 (1/2) (1/4)}=750
50x(15/8+14/8+13/8.....+1/8) = 50x((15x16)/2)/8 = 50x15 = 750
V = .5 x .25 x 50 = 6.25 and in each concrete block, we know that its height increased by 1 consecutively, so the sum of first 15 concrete blocks was 120. Therefore, 6.25 x 120 = 750 :)
V = 1 x (w x l x h) + 2 x(w x l x h) ......+ 15 x (w x l x h) = 1 x (1/4 x 1/2 x 50 ...+ 15 x (1/4 x 1/2 x 50) V = 750 cu.meter
volume of a block = v = 50 0.5 0.25 = 6.25m3 the total volume of 15 steps is = 15v + 14v + 13v + 12v+....+v = v (1+2+3+4+....15) = (15)(16/2)*v = 750m3
The steps are all equal, as they are with the spaces beneath them so we can use the triangle numbers .
The fifteenth triangle number is 120. Found using (1/2 ( n x (n+1))
Each step has a cross section of 1/8 m^2.
Cross Sectional Area of the steps = 120x1/8 = 15 m^2
Volume = Cross Sectional Area x Length = 15 x 50 = 750 m^3
You can count the area from the top where the are is 5 0 × 0 . 2 5 × 0 . 5 = 6 . 2 5
The length 5 0 m and rise . 2 5 m never changes, the tread only changes increased every step in the direction of gravitational force. that is
#
After summing up all the result we get 7 5 0 m 2
volume of 1 block = 50 1/4 1/2 = 25/4
total no. of blocks = 120
total volume = 25/4 * 120 = 750
The length,width is same for all.but height goes .25m up every time. V=50 x .5 x (0.25+0.5+0.75+.......+3.75)
Imagine that the terrace is actually a rectangular prism 50 m x 7.5 m (15/2) x 3.75 m (15/4). The volume of the entire prism would be 1406.25 cubic meters. The volume of concrete on the terrace is 1/2 of 1406.25 cu. m. + the sum of the area of 15 right triangles with base .25 m and height .5 m. 1/2 of 1406.25 is 703.125. Each step (right triangle) has a volume of (.25 m x .5 m x 50 m) / 2 , or 3.125 cu. m. 3.125 cu m x 15 steps = 46.875 cu m. 703.125 cu m + 46.875 cu m = 750 cubic meters
number of staircase would be 1+2+3+... +15 as a staircase cannot be in air. So for 2nd staircase onwards we need stairs below it as well. so adding these using formula for sum of first n natural numbers we get, S=n(n+1)/2 =15(16)/2 =120 therefore total volume V=120 50 (1/4)*(1/2) =750
we have volume of one step=1/4x1/2x50=6.25
since there are 15 steps so the height of each steps is 1/4 more than the last
so we can say , volume of second step is twice the first step and third step is thrice the first and so on......
by this we can write, the total volume of steps as 6.25+2x6.25........+15x6.25
which can be written as 6.25(1+2+3......+15)
which is 6.25x120
which comes to 750
Total no of 50x1/4*1/2 blocks required=1+2+3+...+15=120 So, volume=120x(1/2)x(1/4)x(50)=750
dividing it horizontally, volume of lowset step = (0.5 15) 0.25 50 for last but one step (ie 14th step): (0.5 14) 0.25 50 and so on.. for last step: (0.5 1) 0.25*50
hence total volume = (0.5 0.25 50*(1+2+3---+14+15)) = 750 m3
simple, volume = 50 * 1/2 * ( 1/4 +2/4 + 3/4 + ..........+15/4 ) = 50 * 15 * 16 / ( 2 * 4 * 2 ) =750.
Look at the cross-section of the steps. Clearly, Volume = Area of cross section x 50.
Divide the cross section into 15 parts, each part being a rectangle formed by the extension of the rise of each step. Then,
Area of cross section = (.25x.5) + (.25x2x.5) + (.25x3x.5)....+(.25x15x.5) The sum can easily be calculated by the concept of arithmetic progression. It turns out to be 15.
Thus the volume is 15x50 = 750 cubic meters.
yeah i did it!!!! i just sum up the volume of each step,.. ,..knowing volume by heart is important
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The volume of each block is V = 2 1 × 4 1 × 5 0 = 4 2 5 m 3 . Now, how many blocks are there? Notice that at the first step, there is one block, at the second step, there are two blocks, at the third step three blocks, and so on, until the fifteenth step, which has 15 blocks. This forms an arithmetic progression, with a 1 = 1 , a 1 5 = 1 5 , n = 1 5 and d = 1 . Therefore, the total number of blocks is S = 2 n ( a 1 + a n ) = 2 1 5 ( 1 + 1 5 ) = 1 2 0 . Therefore, the total volume of the blocks is V total = number of blocks × volume of each block = 1 2 0 × 4 2 5 m 3 = 7 5 0 m 3 .