Math, asked by anushka730, 5 months ago

The ratio of the volumes of two spheres is 27 : 64. If the sum of there radii are 28 cm, what is the radius of each of them respectively?​

Answers

Answered by MagicalBeast
17

Let :

  • Radius of first sphere = r
  • Radius of second sphere = R

Given :

  • Ratio of volume of two spheres = 27:64
  • Sum of radius of two spheres = 28 cm

To find :

Radius of each sphere

Formula used :

Volume of sphere = (4/3) π (radius)³

Solution :

Volume of first sphere = (4/3)π r³

Volume of second sphere = (4/3)π R³

Volume of first sphere : Volume of second sphere = (4/3)π r³ : (4/3)π R³

\sf \implies \:\dfrac{ \: Volume \:  of first \:  sphere \:  }{ \: Volume of  \: second  \: sphere}  \: = \:    \dfrac{ \frac{4}{3} \pi r^3 }{ \frac{4}{3} \pi R^3}

\sf \implies \:\dfrac{ \: 27}{64}  \: = \:    \dfrac{  r^3 }{  R^3}

\sf \implies \:\dfrac{ \:  {3}^{3} }{ {4}^{3} }  \: = \:   \bigg(  \dfrac{  r }{  R} \bigg)^3

\sf \implies \:   \bigg(  \dfrac{  r }{  R} \bigg)^3 \:  =  \:  \bigg( \dfrac{ \:  {3} }{ {4} } \bigg)^3 \:

\sf \implies \:     \dfrac{  r }{  R}  \:  =  \:  \dfrac{ \:  {3} }{ {4} }

\sf \implies \:     r  \:  =  \:  \dfrac{ {3} }{ {4} }  R \:  \:  \:  \:  \: equation \: 1

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Sum of radius = r + R

➝ r + R = 28cm

On putting value of r from equation 1 into above equation we get

\sf \implies \:     \dfrac{ {3} }{ {4} }  R \: +   \:  R = 28

\sf \implies \:     \dfrac{ (3 R  \times 1) + (4 \times  R)  }{ {4} }  \:  \:= 28

\sf \implies \:     \dfrac{ 3 R  + 4  R  }{ {4} }  \:    \:= 28

\sf \implies \:     \dfrac{ 7 R     }{ {4} }  \:    \:= 28

\sf \implies \:  R  \: =  \:   \dfrac{ 4  }{ {7} }  \:    \times  28

\sf \implies \:  R  \: =  \:  4 \times 4

\sf \implies \:  R  \: =  \:  16 \: cm

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Now ,

r = (3/4)R

r = (3/4) × 16

r = 48/4

r = 12 cm

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ANSWER :

  • Radius of first sphere (r) = 12 cm
  • Radius of second sphere (R) = 16cm
Answered by Anonymous
30

Question :

The ratio of the volumes of two spheres is 27 : 64. If the sum of there radii are 28 cm, what is the radius of each of them respectively?

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Given :

  • Volume of 2 spheres = 27:64
  • Sum of there radii = 28cm.

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To find :

  • Radius of each sphere.

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Solution:

 \boxed {\underline {\sf Volume \: of \: sphere \: = \dfrac {4}{3} \pi r^3 }}

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 \sf {Ratio \: of \: their \: volumes \:  = V_1 :  V_2 = 27 : 64 }

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 \implies \sf { \dfrac{4}{3} \pi {r_1}^3 = \dfrac{4}{3} \pi {r_2}^3 }

 \implies \sf {{r_1}^3 : {r_2}^3 = 27:64}

\implies \sf {{r_1}^3 : {r_2}^3 = 3:4}

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 \sf {Let,  \: r_1 = 3a ; r_2 = 4a }

 \implies \sf {3a+ 4a = 28 }

 \implies \sf { a = 4}

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 \boxed {\sf Radius (R_1) = 3a = 3(4) = 12cm}

 \boxed {\sf Radius (R_2) = 4a = 4(4) = 16cm}

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