Question 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, 1.0 cm, 1.5 cm, 2.0 cm, ……… as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles?
Let semi circumference of 1st ,2nd ,3rd ,……….. semi be I1, I2, I3, ……..
Semi circumference = ∏r
Therefore, I1, I2, I3 ,i.e. the lengths of the semi-circles are in an A.P.,
Since I1, I2, I3 ,i.e
are in A.P
Therefore,
S13 =?
We know that the sum of n terms of an a A.P. is given by
= 143
Therefore, the length of such spiral of thirteen consecutive semi-circles is 143 cm.
Question 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?
The numbers of logs in rows are in an A.P.
Sol.
Sol.
20, 19, 18…
For this A.P,
a = 20, d = a2 − a1 = 19 − 20 = −1
Let the 200 logs be placed in n rows.
Sn = 200
400 = n (40 − n + 1)
400 = n (41 − n)
400 = 41n − n2
n2 − 41n + 400 = 0
n2 − 16n − 25n + 400 = 0
n (n − 16) −25 (n − 16) = 0
(n − 16) (n − 25) = 0
(n − 16) = 0 or n − 25 = 0
n = 16 or n = 25
an = a + (n − 1)d
a16 = 20 + (16 − 1) (−1)
a16 = 20 − 15
a16 = 5
Similarly,
a25 = 20 + (25 − 1) (−1)
a25 = 20 − 24
= −4
The number of logs in 16th row is 5. The number of logs in 25th row is negative, which is impossible.
Therefore, 200 logs can be placed in 16 rows and the number of logs in the 16th row is 5.