Question

${\sf \huge{\tt {\bf {\pink{\: Q{\frak {\underline {\underline{uestion}}}}}}}}}$
The difference in the areas of two concentric circles is 66 cm² and the radius of the outer circle is 11 cm. What is the radius of the inner circle?

Options
(a) 8 cm
(b) 9 cm
(c) 10 cm
(d) 7 cm​

Answer 1

Option (c) 10 cm is the correct answer.

Explanation :

${\boxed{\boxed{\begin{array}{cc} \underline{\bf \: \to \:given : } \\ \\ \hookrightarrow \: \sf \:radius \: of \: outer \: circle \: \: R = 11 \: cm \\ \\ \hookrightarrow \: \sf \:The \: difference \: in \: the \: areas \: of \: two \\ \sf \: concentric \: \: circles \: is \: \: \Delta\:A = 66 \: {cm}^{2} \\ \\ \\ \pink{ \underline{ \sf \: we \: have \: to \: find : \: }}\\ \\ \boxed{ \blue{ \hookrightarrow \: \sf \:radius \: of \: the \: inner \: circle = r}} \\\\\end{array}}}}$

${\boxed{\boxed{\begin{array}{cc} \red{ \underline{ \bf \: solution}} \\ \\ \bf \: let : \\ \\ \leadsto \: \sf\: area \: of \: outer \: circle \: \: A_{outer} = \pi {R}^{2} \\ \\ \sf \: \leadsto \: area \: of \: inner \: circle \: \: A_{inner} = \pi {r}^{2} \\ \\ \red{ \sf \: according \: to \: the \: question \: } \\ \\ \sf \: A_{outer} - A_{inner} = \Delta\:A \\ \\ \sf \: \implies \: \pi {R}^{2} - \pi {r}^{2} = 66 \\ \\ \sf \: \implies \:\pi( {R}^{2} - {r}^{2}) = 66 \\ \\ \sf \: \implies \: \frac{22}{7} ( {R}^{2} - {r}^{2} ) = 66 \\ \\ \sf \: \implies \: {R}^{2} - {r}^{2} = \frac{66 \times 7}{22} \\ \\ \sf \: \implies \: {r}^{2} = {R}^{2} - \frac{66 \times 7}{22} \\ \\ \sf \: \implies \: {r}^{2} = {(11)}^{2} - \frac{ \cancel{66} {}^{ \: \: 3} \times 7}{ \cancel{22}} \\ \\ \sf \: \implies \: {r}^{2} = 121 - 3 \times 7 \\ \\ \sf \: \implies \: {r}^{2} = 121 - 21 \\ \\ \sf \: \implies \: {r}^{2} = 100 \\ \\ \sf \: \implies \:r = \sqrt{100} \\ \\ \sf \: \implies \:r = 10 \: cm \\ \\ \\ \orange{ \boxed{ \therefore \: \sf \: radius \: of \: inner \: circle \: is \: \: r = 10 \: cm}}\end{array}}}}$

Answer 2

Answer:

• Option (c) 10 cm is correct.

Explanation:

Given information,

The difference in the areas of two concentric circles is 66 cm² and the radius of the outer circle is 11 cm. What is the radius of the inner circle?

• Difference in the areas of two concentric circles = 66 cm²
• Radius of outer circle (R) = 11 cm
• Radius of inner circle (r) = ?

Given options,

• (a) 8 cm
• (b) 9 cm
• (c) 10 cm
• (d) 7 cm

Using formula,

✪ Area of circle = πr² ✪

Where,

• π denotes pi
• r denotes radius of circle

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Here, we are given that the difference in the areas of two concentric circles is 66 cm². So, area of outer circle - area of inner circle = 66 cm². Therefore,

➻ Area(outer circ) - Area(inner circ) = 66

Using formula of area of circle; As outer circle is big so we will take it's radius as R and inner circle is small so we will take it's radius as r.

➻ πR² - πr² = 66

➻ π(R² - r²) = 66

We have,

• π = 22/7
• R = 11 cm
• r = ?

Putting all values,

➻ 22/7 × (11² - r²) = 66

➻ 22/7 × (121 - r²) = 66

➻ 121 - r² = 66 ÷ 22/7

➻ 121 - r² = 66 × 7/22

➻ 121 - r² = 3 × 7/1

➻ 121 - r² = 3 × 7

➻ 121 - r² = 21

➻ - r² = 21 - 121

➻ - r² = - 100

➻ r² = 100

➻ r = √100

➻ r = √(10 × 10)

➻ r = 10

• Hence, radius of inner circle is 10 cm. So, option (c) 10 cm is correct.

Know more,

• Circles having same center are known as concentric circle.

Some formulae,

• Area of circle = πr²
• Area of square = (side)²
• Area of rectangle = L × B
• Area of trapezium = ½(a + b)h
• Area of rhombus = ½(d₁ × d₂)
• Area of equilateral ∆ = √3/4(side)²

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