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 (1/4) m and a tread of (1/2)m. Calculate the total volume of concrete required to build the terrace.

 

 cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.4 optional exercise

 

5. 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 (1/4) m and a tread of (1/2)m.   Calculate the total volume of concrete required to build the terrace.

 

thinking of each step as a cuboid with length =50m, breadth =(1/2)m

and  height (1/4)m for the first step with increase of (1/4)m for each further step

volume for  first step

V1 = length * breadth * height =50*(1/2)*(1/4)

 

for second step 

 length =50m, breadth =(1/2)m, height =(1/4)+(1/4) =(1/2)m

volume for  second step

V2 = length * breadth * height =50*(1/2)*(1/2)

 

for third step 

 length =50m, breadth =(1/2)m, height =(1/4)+(1/4)+(1/4) =(3/4)m

volume for  second step

V2 = length * breadth * height =50*(1/2)*(3/4)

 

and so on

 

so total volume

= [50*(1/2)*(1/4)]+[50*(1/2)*(1/2)]+[50*(1/2)*(3/4)] +... (15 terms)

=50*(1/2) { (1/4) +(1/2) +(3/4) + ....(15 terms) }

=25 *{ (1/4) +(1/2) +(3/4) + ....(15 terms) } ,

using sum of n=15 terms of an AP with first term a=(1/4),  d =(1/4) for { ...}

Sn = (n/2)*[ 2a + (n-1)d ]

total volume = 25 *{ (15/2) *[2(1/4) +(15-1)(1/4) ] }

=25* { (15/2) [ (1/2)+ (14/4)  ] }

=25* {15/2 [(1/2) +(7/2)}

=25* {(15/2)*(4)}

=25*15*2=750 cubic metres

 

 cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.3

 exercise 5.3

 20. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato,and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

the distance covered to bring the first potato to the bucket = 5+5=10m

 

After that ,the distance covered for bringing the second potato to the bucket

=5 +3 + 3 +5 =16m

 

After that ,the distance covered for bringing the third potato to the bucket

=5 +3 + 3 +3+ 3 +5 =22m 

and so on for the 10 potatoes

 

so total distance = 10 +16 +22 + ...  (10 terms )

 

using sum of n=10 terms of an AP with first term a=10 , d=t2-t1=16-10=6

Sn = (n/2)*[ 2a + (n-1)d ]

 

total distance = 10 +16 +22 + ...  (10 terms )

= (10/2)*[2(10)+(10-1)(6)]

=5*74 =370m

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ncert cbse 10th mathematics

chapter 5  arithmetic progressions 

exercise 5.4 optional exercise

Which term of the AP : 121, 117, 113, . . ., is its first negative term? 

solution

2. The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.

solution

 3. A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and last rungs are [ 2 and(1/2) ]m apart, what is the length of the wood required for the rungs?

solution

 4. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.

solution

5. 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 (1/4) m and a tread of (1/2)m.   Calculate the total volume of concrete required to build the terrace.

 solution

 

chapter 5 arithmetic progressions, exercise 5.3

 exercise 5.3

 20. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato,and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

 solution

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