Question

A spiral is made up of successive semicircles with centres alternately at a and b starting with center at a of radii 0.5 cm 1cm 1.5cm ...What is the total length of such a spiral made up of 13 consecutive semicircles

Answer 1

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, 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?


Answers :-



Here length means circumference of the semi circles l(1),l(2),l(3),l(4)..........

Circumference of semi

circle =πr l(1)=0.5π l(2)=1.0π

l(3)=1.5π And so on So the

AP is 0.5π,1.0π,1.5π.....

now , d = 1.0π-0.5π=0.5π

Sn =n/2{2a+(n-1 )d}

Sn=13/2{2(0.5π)+(13-1)

(0.5π)} Sn=13/2{1π+(12)

(0.5π) Sn=13/2{1π +6π}

Sn=13/2(7π) Sn = 13/2 ×7×22/7 Sn = 13/2×22 Sn=

13 ×11 Sn=143 (Ans )



Here,

Semi-perimeter of circle = πr

I1 = π(0.5)

I2 = π(1) = π cm

I3 = π(1.5) =

Therefore, I1, I2, I3 ,i.e. the lengths of the semi-circles are in an A.P.,

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 will be 143 cm .

Answer 2

$\Large{\underline{\underline{\bf{Given:-}}}}$

A spiral is made up of successive semicircles with centres alternately at a and b starting with center at a of radii 0.5 cm 1cm 1.5cm.

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$\Large{\underline{\underline{\bf{To \: Find:-}}}}$

The total length of such a spiral made up of 13 consecutive semicircles

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$\Large{\underline{\underline{\bf{Formula\: used:-}}}}$

$s_n = \frac{n}{2}[ 2a + (n - 1)d]$

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$\Large{\underline{\underline{\bf{Solution:-}}}}$

Length of semi-circle=

$\frac{circumference \: of \: a \: circle}{2} \\ \\ </strong><strong>⟹</strong><strong> = \frac{2\pi \: r}{2} \\ \\ </strong><strong>⟹</strong><strong> = \pi \: r$

Lengthofsemi-circleofradii0.5cm = (0.5)cm

Length of semi-circle of radii 1.0cm = (1.0)cm

Lengthofsemi-circleofradii 1.5cm = (1.5)cm

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$\Large{\underline{\underline{\bf{Therefore\:,\: sequence\: of \: the \: form:-}}}}$

π(0.5), π(1.0), π(1.5)...13terms

{There are total of thirteen semi–circles}.

To find total length of the spiral,we need to find sum of the sequence π(0.5), π(1.0),

π(1.5)...13terms

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

⇒ Total length of spiral = π(0.5 + 1.0 + 1.5)...13 terms......( i )

◕Sequence 0.5, 1.0 ,1.5...13 terms is an arithmetic progression.

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$\Large{\underline{\underline{\bf{Let\: us \: find \: the \:Sum\: of \: this\: sequence:-}}}}$

$⟹a = 0.5 \\ \\⟹ d = 1.0 - 0.5 = 0.5 \\ \\⟹ n = 13$

$\Large{\underline{\underline{\bf{By \: using \: the \: above \: formula:-}}}}$

$⟹s_n = \frac{n}{2}[2a + (n - 1)d] \\ \\⟹ s_{13} \: = \frac{13}{2} [1 + (13 - 1)0.5 ]\\ \\⟹ s_{13} = 6.5(1 + 6) \\ \\ ⟹s_{13} = 6 .5 \times 7 \\ \\ ⟹s_{13} = 45.5$

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◕Therefore, 0.5 + 1.0 + 1.5 + 2.0...13 terms = 45.5

Putting this in equation ( i ) ,we get

◕Total length of spiral = (0.5 + 1.5 + 2.0 +...13terms) = (45.5)=143 cm

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${\huge\boxed{Total \: length \: = \: 143\: cm}}$

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$\Large{\underline{\underline{\bf{Thanks}}}}$