Question

39. The width of a ring is 6 cm and the area of the inner circle is 616 cm2. Find the circumference of the outer circle.​

Answer 1

Explanation:

Given,

πr² = 616

r² = 616×7/22

r ² = 28×7

r = 14

Therefore, Circumference = 2πr = 2× 22/7 ×14 = 88 cm

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Answer 2

★ The circumference of the outer circle is 880/7 cm.

Step-by-step explanation

Analysis -

‎ ‎ ‎ ‎ ‎ ‎ ‎In the question, it has been given that the width of a ring is 6 cm. Since a ring is in the structure of concentric circles, the area of its inner circle is 616 cm². We've been asked to calculate the circumference (perimeter) of the outer circle.

Solution -

‎ ‎ ‎ ‎ ‎ ‎ ‎First, we should find the length of radius of inner circle (r) by using the formula of area of circle because, the radius of outer circle (R) is dependent upon the radius of inner circle. Let us plug in the values in the formula and solve it.

$\dashrightarrow{\sf\pmb{Area_{(circle)}=\pi r^2}}\\ \\ \dashrightarrow{\sf{616=\dfrac{22}{7} \times r^2}}\\ \\ \dashrightarrow{\sf{616(7)=22\times r^2}} \\ \\ \dashrightarrow{\sf{4312=22 \times r^2}}\\ \\ \dashrightarrow{\sf{\dfrac{\cancel{4312}}{\cancel{22}}=r^2}}\\ \\ \dashrightarrow{\sf{196=r^2}}\\ \\ \dashrightarrow{\sf{\sqrt{196}=r}}\\ \\ \dashrightarrow\underline{\boxed{\sf\pmb{14\:cm=r}}}$

We've found the length of radius of inner circle (r). Now let us know what makes the radius of outer circle (R) to be a dependent character.

Have a look at the diagram attached! You may observe that, it has been marked 6 cm as the width of the ring (w). This value doesn't equals the radius of outer circle (R). The actual value of R equals the sum of radius of inner circle (r) and width of the ring (w), i.e.,

‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎R = r + w

On inserting the values,

$\twoheadrightarrow{\sf{R=14+6}}\\ \\ \twoheadrightarrow\underline{\boxed{\sf\pmb{R=20\:cm}}}$

The length of radius of outer circle is 20 cm. So, finally, let us use the formula of circumference of circle to find the perimeter of outer circle.

$\:\:\:\sf{\pmb{Circumference_{(circle)}=2\pi R \:units}}$

Plugging in the values,

$\dashrightarrow{\sf{2\times \dfrac{22}{7}\times 20}}\\ \\ \dashrightarrow{\sf{\dfrac{2\times 22\times 20}{7}}}\\ \\ \dashrightarrow{\sf{\dfrac{44\times 20}{7}}}\\ \\ \dashrightarrow\underline{\boxed{\frak{\pmb{\purple{\dfrac{880}{7}\:cm}}}}}$

Therefore, the circumference of the outer circle is 880/7 cm.

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