Question

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 semi circles. PLEASE SAY CORRECT ANSWER​

Answer 1

Total length = 143cm

Step-by-step explanation:

Take π = 22/7

circumference of semi circle = 1/2 × 2πr = πr

so, multiply the given radii by π , i.e.,

circumference of 1st semi circle = π × 0.5

2nd semi circle = π × 1.0

3rd semi circle = π × 1.5 continue it

likewise, for 13th semi circle = π × 6.5

Now, this series is : π × 0.5 , π×1.0, π× 1.5,.....π×6.5

so take π common , this becomes

π ( 0.5, 1.0, 1.5, 2.0 .....6.5)

π S ( S = sum of the series)

0.5, 1.0, 1.5, 2.0....6.5

This series is in A.P where the common difference c. d = a2 - a1 = a3 - a2 = 0.5

use this formula S = n/2 (a + l) [where a is the first term, l is the last term], n = 13, a= 0.5 , l = 6.5

Put the values in the formula,

S = 13/2 ( 0.5 + 6.5)

= 13/2 × 7

= 91/2

= 45.5

Now, length of spiral = Sum of series

= π × S

= π × 45.5

= 22/7 × 45.5

= 22 × 6.5

= 143

Length of the spiral is 143 cm.