Question

A trigonometric proving question

Answer 1

Answer:

Correct Question -

The circumference of two circle are in the ratio 2 : 3. Find the ratio of their areas.

Given -

Ratio of their circumference = 2:3

To find -

Ratio of their areas.

Formula used -

Circumference of circle

Area of circle.

Solution -

In the question, we are provided, with the ratios of the circumference of 2 circles, and we need to find the ratio of area of those circle, for that first we will use the formula of circumference of a circle, then we will use the formula of area of circles. We will be writing 1 equation in it too.

So -

Let the circumference of 2 circles be c1 and c2

According to question -

c1 : c2

Circumference of circle = 2πr

where -

π = $\tt\dfrac{22}{7}$

r = radius

On substituting the values -

c1 : c2 = 2 : 3

2πr1 : 2πr2 = 2 : 3

$\tt\dfrac{2\pi\:r\:1}{2\pi\:r\:2}$ = $\tt\dfrac{2}{3}$

$\tt\dfrac{r1}{r2}$ = $\tt\dfrac{2}{3}$$\longrightarrow$ [Equation 1]

Now -

Let the areas of both the circles be A1 and A2

Area of circle = πr²

So -

Area of both circles = πr1² : πr2²

On substituting the values -

A1 : A2 = πr1² : πr2²

$\tt\dfrac{A1}{A2}$ = $\tt\dfrac{(\pi\:r1)}{(\pi\:r2)}^{2}$

$\tt\dfrac{A1}{A2}$ = $\tt\dfrac{(r1)}{(r2)}^{2}$

$\tt\dfrac{A1}{A2}$ = $\tt\dfrac{(2)}{(3)}^{2}$ [From equation 1]

So -

$\tt\dfrac{A1}{A2}$ = $\tt\dfrac{4}{9}$

$\therefore$ The ratio of their areas is 4 : 9

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Answer 2

Answer:

Answer:

Correct Question -

The circumference of two circle are in the ratio 2 : 3. Find the ratio of their areas.

Given -

Ratio of their circumference = 2:3

To find -

Ratio of their areas.

Formula used -

Circumference of circle

Area of circle.

Solution -

In the question, we are provided, with the ratios of the circumference of 2 circles, and we need to find the ratio of area of those circle, for that first we will use the formula of circumference of a circle, then we will use the formula of area of circles. We will be writing 1 equation in it too.

So -

Let the circumference of 2 circles be c1 and c2

According to question -

c1 : c2

Circumference of circle = 2πr

where -

π = \tt\dfrac{22}{7}

7

22

r = radius

On substituting the values -

c1 : c2 = 2 : 3

2πr1 : 2πr2 = 2 : 3

\tt\dfrac{2\pi\:r\:1}{2\pi\:r\:2}

2πr2

2πr1

= \tt\dfrac{2}{3}

3

2

\tt\dfrac{r1}{r2}

r2

r1

= \tt\dfrac{2}{3}

3

2

\longrightarrow⟶ [Equation 1]

Now -

Let the areas of both the circles be A1 and A2

Area of circle = πr²

So -

Area of both circles = πr1² : πr2²

On substituting the values -

A1 : A2 = πr1² : πr2²

\tt\dfrac{A1}{A2}

A2

A1

= \tt\dfrac{(\pi\:r1)}{(\pi\:r2)}^{2}

(πr2)

(πr1)

2

\tt\dfrac{A1}{A2}

A2

A1

= \tt\dfrac{(r1)}{(r2)}^{2}

(r2)

(r1)

2

\tt\dfrac{A1}{A2}

A2

A1

= \tt\dfrac{(2)}{(3)}^{2}

(3)

(2)

2

[From equation 1]

So -

\tt\dfrac{A1}{A2}

A2

A1

= \tt\dfrac{4}{9}

9

4

\therefore∴ The ratio of their areas is 4 : 9

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