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4.* If the ratio of the curved surface areas of a solid cone and a solid right circular cylinderN having same base radii and same height is 5:8, then let us determine the ratio of their baseradii and heights.​

Answer»

Let the same base radii be r units and same height be H units

Given :

Ratio of CSAs of cone and cylinder with same height and base = 5 : 8

  • CSA of cone = πrl sq.units
  • CSA of cylinder = 2πrh sq.units

⇒ πrl / 2πrh = 5 / 8

⇒ l / 2h = 5 / 8

We KNOW that

  • l = √( r² + h² )

⇒ √( r² + h² ) / 2h = 5 / 8

Squaring on both sides

⇒ [ √(r² + h²) / 2h ]² = ( 5 / 8 )²

⇒ ( r² + h² ) / 4h² = 25 / 64

⇒ 64r² + 64h² = 100h²

⇒ 64r² = 100h² - 64h²

⇒ 64r² = 36h²

⇒ r² / h² = 36 / 64

⇒ ( r / h )² = 36 / 64

Taking square root on both sides

⇒ √( r / h )² = √( 36 / 64 )

⇒ r / h = 6 / 8

⇒ r / h = 3 / 4

⇒ r : h = 3 : 4

Therefore the ratio of base RADIUS and height is 3 : 4.


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