If the areas of two circles are in the ratio 25:36 then the ratio between their circumferences is

If the ratio of areas of two circles is $$16 : 25$$, the ratio of their circumference is

• A

  $$25 : 16$$

• B

  $$5 : 4$$

• C

  $$4 : 5$$

• D

  $$3 : 5$$

Then, their circumferences are $$2\pi { r }_{ 1 }$$ & $$2\pi { r }_{ 2 }$$, respectively.

So, their ratio=$$2\pi { r }_{ 1 }:2\pi { r }_{ 2 }={ r }_{ 1 }:{ r }_{ 2 }$$.

Again, the areas of the circles are $$\pi { \left( { r }_{ 1 } \right)  }^{ 2 }$$ & $$\pi { \left( { r }_{ 2 } \right)  }^{ 2 }$$.

Then, their ratio $$=\pi { \left( { r }_{ 1 } \right)  }^{ 2 }:\pi { \left( { r }_{ 2 } \right)  }^{ 2 }={ \left( { r }_{ 1 } \right)  }^{ 2 }:{ \left( { r }_{ 2 } \right)  }^{ 2 }=16:25$$     ...(given).

$${ \therefore \left( { r }_{ 1 } \right)  }:{ \left( { r }_{ 2 } \right)  }=\sqrt { 16:25 } =\pm \left( 4:5 \right) $$

We reject the negative value of r's since the radius is a length.

$${ \therefore \quad \left( { r }_{ 1 } \right)  }:{ \left( { r }_{ 2 } \right)  }=\left( 4:5 \right) $$.

Ratio of areas of circles =25:36=25/36Radius of first circle = πr^2Radius of second circle = πR^2πr^2/πR^2 = 25/36r^2/R^2 = 25/36(r/R)^2 = 25/36r/R = √25/36r/R =5/6Ratio of circumferences=2πr/2πR=r/R=5/6

Therefore, ratio of circumferences = 5:6