200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on. In how many rows are the 200 logs placed and how many logs are in the top row?

cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.3

 

 

19.

 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on . In how many rows are the 200 logs placed and how many logs are in the top row?

 

using arithmetic progressions AP

 t1 = 20

t2 = 19

t3=18

a =20 

d=t2 - t1 = 19 - 20 = (-1)

total number of logs is 200

so S(n) = 200 , n = ?

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

 

200 = (n/2)*[ 2(20) + (n-1)(-1) ]

200 =  (n/2)*[ 40 - n +1 ]

200= (n/2)*[ 41 - n  ]

200*2 = n[41-n]

400 = 41n -(n^2)

 

(n^2) - 41n +400 =0

 

(n-16) (n-25) =0

n = 16, n=25

if n = 16

tn = a+(n-1)d

t(16) = 20 + (16-1)(-1) = 20+(15)(-1) = 20-15 = 5 logs in the top row

if n = 25

tn = a+(n-1)d

t(25) = 20 + (25-1)(-1) = 20+(24)(-1) = 20-24= (-4) logs which is not possible.

 

So number of rows is 16, and number of logs in the top row is 5.

18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . .  What is the total length of such a spiral made up of thirteen consecutive semicircles.

 

length of arc of semi circle   = pi*r

Total length of spiral = pi*0.5 +pi*1.0+pi*1.5 + .... (13 terms)

=pi [0.5 + 1.0 +1.5 + ...  (13 terms) ]

using formula for sum of n terms of an AP

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

with n =13, a=0.5, d=1.0-0.5 = 0.5

 

Total length of spiral =pi { (13/2) [ 2(0.5)+(13-1)(0.5) ] }

=pi  { (13/2) [1.0 +12*0.5 ] }

=pi { (13/2) [1.0 +6.0 ] }

=pi {(13/2)[7]}

=(22/7) {(13/2)[7]}

=11*13

=143 cm

 

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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

 

19.

 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on . In how many rows are the 200 logs placed and how many logs are in the top row?

solution 

 

 18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . .  What is the total length of such a spiral made up of thirteen consecutive semicircles.

solution 

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