Q : 18       A spiral is made up of successive semicircles, with centres alternately at \small A and \small B​​​​​​, starting with centre at \small A, of radii \small 0.5\hspace {1mm}cm,1.0\hspace {1mm}cm,1.5\hspace {1mm}cm,2.0\hspace {1mm}cm,...  as shown in Fig. \small 5.4. What is the total length of such a spiral made up of thirteen  consecutive semicircles? (Take   \pi =\frac{22}{7}  )

               

                [Hint : Length of successive semicircles is  \small l_1,l_2,l_3,l_4,...  with centres at  \small A,B,A,B,..., respectively.]

Answers (1)

From the above given figure

Circumference of 1st semicircle l_1 = \pi r_1 = 0.5\pi

Similarly,

Circumference of 2nd semicircle l_2 = \pi r_2 = \pi

Circumference of 3rd semicircle l_3 = \pi r_3 = 1.5\pi

It is clear that this is an AP with a = 0.5\pi \ and \ d = 0.5\pi

Now, sum of length of 13 such semicircles is given by

S_{13} = \frac{13}{2}\left \{ 2\times 0.5\pi + (13-1)0.5\pi\right \}
S_{13} = \frac{13}{2}\left ( \pi+6\pi \right )
S_{13} = \frac{13}{2}\times 7\pi
S_{13} = \frac{91\pi}{2} = \frac{91}{2}\times \frac{22}{7}=143
Therefore, sum of length of 13 such semicircles is 143 cm 

Most Viewed Questions

Preparation Products

Knockout NEET May 2024

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 40000/-
Buy Now
Knockout NEET May 2025

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 45000/-
Buy Now
NEET Foundation + Knockout NEET 2024

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 54999/- ₹ 42499/-
Buy Now
Knockout JEE Main April 2021 (One Month)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 14000/- ₹ 4999/-
Buy Now
Knockout NEET Aug 2021 (One Month)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 14000/- ₹ 4999/-
Buy Now
Boost your Preparation for JEE Main with our Foundation Course
 
Exams
Articles
Questions