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(B) The radii of the $14$ discs form an arithmetic progression: $1, 2, 3, \dots, 14$.The circumference of a disc with radius $r$ is given by $C = 2\pi

Vedclass · Accepted Answer

$660$

Vedclass · Answer

(B) The radii of the $14$ discs form an arithmetic progression: $1, 2, 3, \dots, 14$.The circumference of a disc with radius $r$ is given by $C = 2\pi r$.The sum of the circumferences of all $14$ discs is $S = 2\pi(1) + 2\pi(2) + 2\pi(3) + \dots + 2\pi(14)$.$S = 2\pi(1 + 2 + 3 + \dots + 14)$.Using the sum formula for the first $n$ natural numbers, $S_n = \frac{n(n+1)}{2}$, where $n = 14$:$S = 2\pi 	imes \frac{14(14+1)}{2} = 2\pi 	imes \frac{14 	imes 15}{2} = 2\pi 	imes 105 = 210\pi$.Taking $\pi \approx \frac{22}{7}$:$S = 210 	imes \frac{22}{7} = 30 	imes 22 = 660\, cm$.

$A$ plastic toy consists of $14$ circular discs of different radii. The radius of the smallest disc is $1\, cm$, the radius of the next bigger disc is $2\, cm$, and so on, such that the radius of the biggest disc is $14\, cm$. Find the sum of the circumferences of all the discs in $cm$.

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