Question

The ratio of the circumference of two circles is 2: 3 and the difference in length of their half-diameter is 2 cm. Find the length of the diameter of both circles.

Answer 1

Given:

• Ratio of circumference of two circle is 2:3.
• The difference in their half of diameter is 2cm.

To Find:

• Length of diameters of two circles.

Answer:

Let the radius of larger circle be R and smaller second circle be r .

Now , circumference of circle is given by 2πr.

Atq ,

⇒ 2πr:2πR = 2 : 3

⇒ 2πr/2πR = 2/3 .

⇒r/R = 2/3 .

☞ r = 2R/3 . ................(i)

Now according to second condition ,

• Difference of half of their diameters is 2cm .

And , half of diameter is radius .

Hence this implies that the difference of Radius of two circles is 2cm.

⇒R - r = 2 cm.

⇒ R - 2R/3 = 2cm. [from .......(i)]

⇒3R - 2R / 3 = 2cm.

⇒ R/3 = 2cm.

⇒R = 2cm × 3 .

☞ R = 6cm .

Hence radius of larger circle is 6cm.

Putting this value in (i) ;

⇒r = 2R/3 .

⇒r = 2×6cm/3.

☞ r = 4cm.

Hence radius of smaller circle is 4cm.

Answer 2

$\large\sf \red {\underline {\underline {Given:-}}}$

• Ratio of the circumference of two circles is 2:3

• Difference in the length of their half-diameter is 2cm

$\large\sf \blue {\underline {\underline {To\:Find:-}}}$

• The length of diameter of both circles.

$\large\sf \green {\underline {\underline {Solution:-}}}$

Let the radius of larger circle be ' R ' and the smaller circle be ' r '

As we know that the circumference of a circle is given by the formula 2πr

Condition 1:-

$\rightarrow \sf(2\pi R)\: : \:(2\pi r) = 2 \: : \: 3$

$\rightarrow \sf \dfrac{2\pi r}{2\pi R} = \dfrac{2}{3}$

$\rightarrow \sf \dfrac{ r}{R} = \dfrac{2}{3}$

$\rightarrow \sf \dfrac{ r}{ R} = \dfrac{2}{3}$

$\rightarrow \rm { r} = \dfrac{R 2}{3} .....(1)$

Condition 2:-

☞︎︎︎ Difference in the length of their half-diameter is 2cm

Here, half-diameter = radius

Therefore:-

$\sf \rightarrow R - r = 2......(ii)$

Substituting equation(i) in (ii)

$\sf \rightarrow R - \dfrac{2 R}{3} = 2$

$\sf \rightarrow \dfrac{3R - 2 R}{3} = 2$

$\sf \rightarrow R = 2 \times 3$

$\sf \rightarrow R = 6$

Substitute R = 6 in equation (ii)

$\sf \rightarrow R - r = 2$

$\sf \rightarrow 6 - r = 2$

$\sf \rightarrow r = 4$

Hence;

$\large \boxed{ \sf \pink{ Radius \: of \: larger \: circle= 6cm}}$

$\large \boxed{ \sf \purple{ Radius \: of \: smaller \: circle= 4cm}}$