Question

If a wire of 440 m length is moulded in the form of a circle and a square turn -by- turn , find the ratio of the area of the circle to that of square .​

Answer 1

Answer:

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Step-by-step explanation:

$Given , \: length \: of \: wire = 440m$

$which \: means \: the \: boundary \\ \: of \: Circle \: and \: Square.$

$Circumference \: of \: circle = 440m$

$=>2πr = 440m$

$=> πr = 220m$

$=> r = \frac{22 \times 7}{22} m$

$= r = 70m$

$Perimeter \: of \: Square= 440m$

$=> 4a = 440m$

$=> a = 110m$

$Area \: of \: Square = a²$

$= (110)² \: m²$

$= 12100m²$

$Area \: of \: Circle = πr²$

$= \frac{22}{7} ×70 ×70 \: m²$

$= 22×10×70 \: m²$

$= 15400 m²$

$Now, Calculating \: Ratio \: of \: the \\ area \: of \: the \: \\ circle \: to \: that \: of \: square :$

$= \frac{ 15400}{12100}$

$= \frac{4}{2} =\frac{2}{1}$

$= 2:1$

$Therefore, the \: ratio \: of \: the \\ \: area \: of \: the \: circle \\ \: to \: that \: of \: square \: is \: 2:1.$

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Answer 2

Given :

• Length of a wire = 440 m
• The wire is moulded in the form of a circle and a square turn by turn.

To find :

• Ratio of the area of the circle to that of square

Concept :

The length of wire 440 m will be equal to the circumference of circle and perimeter of square.

So, by using the formula of circumference find the value of radius of the circle and by using the formula of perimeter of square find the side of the square.

Then substitute the values in the formula of the areas and then calculate their ratio.

Formulas to be used :-

→ Formula of Circumference of circle :-

$\boxed{ \sf \pmb{Circumference = 2\pi r}}$

→ Formula to calculate area of circle :-

$\boxed{ \sf \pmb{Area \: of \: circle = \pi {r}^{2} }}$

→ Formula of perimeter of the square :-

$\boxed{ \sf \pmb{Perimeter \: of \: square = 4 \times side }}$

→ Formula to calculate area of square :-

$\boxed{ \sf \pmb{Area \: of \: square = {side}^{2}}}$

Solution :

Circumference of the circle = 440 m

Radius of the circle :-

$\\ \dashrightarrow \sf \quad Circumference = 2\pi r$

$\\ \dashrightarrow \sf \quad 440 = 2 \times \dfrac{22}{7} \times r$

$\\ \dashrightarrow \sf \quad 440 = \dfrac{44}{7} \times r$

$\\ \dashrightarrow \sf \quad 440 \times \dfrac{7}{44} = r$

$\\ \dashrightarrow \sf \quad \not440 \times \dfrac{7}{ \not44} = r$

$\\ \dashrightarrow \sf \quad 10 \times 7 = r$

$\\ \dashrightarrow \sf \quad 70 = r$

• Radius of the circle = 70 m

Area of circle :-

$\\ \dashrightarrow \sf \quad Area \: of \: circle = \pi {r}^{2}$

$\\ \dashrightarrow \sf \quad Area \: of \: circle = \dfrac{22}{7} \times ( {70})^{2}$

$\\ \dashrightarrow \sf \quad Area \: of \: circle = \dfrac{22}{7} \times 70 \times 70$

$\\ \dashrightarrow \sf \quad Area \: of \: circle = \dfrac{22}{ \not7} \times \not 70 \times 70$

$\\ \dashrightarrow \sf \quad Area \: of \: circle = 22 \times 10 \times 70$

$\\ \dashrightarrow \sf \quad Area \: of \: circle = 15400$

• Area of circle = 15400 m²

Perimeter of square = 440 m

Side of the square :-

$\\ \dashrightarrow \sf \quad Perimeter \: of \: square = 4 \times side$

$\\ \dashrightarrow \sf \quad 440 = 4 \times side$

$\\ \dashrightarrow \sf \quad \dfrac{440}{4} = side$

$\\ \dashrightarrow \sf \quad 110 = side$

• Side of the square = 110 m

Area of square :-

$\\ \dashrightarrow \sf \quad Area \: of \: square = {side}^{2}$

$\\ \dashrightarrow \sf \quad Area \: of \: square = {110}^{2}$

$\\ \dashrightarrow \sf \quad Area \: of \: square = 12100$

• Area of square = 12100 m²

Ratio of the area of the circle to that of square :-

$\\ \sf Ratio = \dfrac{Area \: of \: circle}{Area \: of \: square}$

$\\ \dashrightarrow \quad \sf \dfrac{15400}{12100}$

$\\ \dashrightarrow \quad \sf \dfrac{154 \not0 \not0}{121 \not0 \not0}$

$\\ \dashrightarrow \quad \sf \dfrac{154}{121}$

$\\ \dashrightarrow \quad \sf \dfrac{ \not154}{ \not121}$

$\\ \dashrightarrow \quad \sf \dfrac{14}{11}$

• Ratio of the area of the circle to that of square = 14 : 11