Question

12.
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, ...
A B
35
22
semicircles? (Take =
7
la
(Hint: Length of successive semicircles is lı, 12, 13, 14, ... with centres at A, B, A, B, ...,
respectively.]
as shown in Fig. 5.4. What is the total length of
such a spiral made up of thirteen consecutive​

Answer 1

Answer:

Total length of such a spiral made up of thirteen consecutive semicircles = 143 cm

Step-by-step explanation:

I believe your Question was,

"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?"

Now,

The spiral is made up of semicircles alternating at A and B.(Do refer the above image)

So,

we must find the total length of all the spirals, since the spirals are made up of semicircles, we must find their circumference.

Now, we know that semicircles have half the circumference of a circle

Thus,

Circumference of Semicircle = (2πr)/2 = πr

Circumference of 1st Semicircle = π × 0.5

Circumference of 2nd Semicircle = π × 1

Circumference of 3rd Semicircle = π × 1.5

.

.

Circumference of 13th Semicircle = π × 6.5

Thus,

the total length of the spiral = 0.5π + 1π + 1.5π......6.5π

Taking π common we get

π(0.5 + 1 + 1.5........6.5)

Hence it forms an AP

a = 0.5

d = 1 - 0.5 = 0.5

Now, we must find their sum,

Sn = (n/2)[a + l]

here,

a = 0.5

d = 0.5

n = 13 (because 13 semicircles are there)

l (last term) = 6.5

S(13) = (13/2)[0.5 + 6.5]

S(13) = (13/2)(7)

S(13) = 91/2

S(13) = 45.5

Thus,

Total length of the spiral = π(45.5)

Taking π = 22/7

Total length = (22/7)(45.5)

= 22 × 6.5

= 143 cm

∴ Total length of such a spiral made up of thirteen consecutive semicircles = 143 cm

Hope it helped and you understood it........All the best