Question

A piece of paper is in the form of a sector, making an angle . The paper is rolled to form a right circular cone of radius 5 cm and height 12 cm. Then the value of  is (in radians)_________​

Answer 1

Answer:

Lateral or slanting height of cone L

      = radius of the circle from which the sector is cut  

 L   = 12 cm

Arc length of the 120° sector = 2π * radius * 120°/360° = 8 π cm

Circumference of the base circle of cone = arc length = 8π cm = 2 π R

 So,   R  = radius of the circle of base of cone = 4 cm

     Area of base of cone = πR² = 16 π cm²

   Altitude or Height of cone = H = √(L² - R²) = √(12² - 4²) = √128 = 8√2 cm

Volume or capacity of Cone  =  1/3 * base area *  Altitude  

                 = 1/3 * 16 π * 8 √2 = 128√2π / 3 cm³

Answer 2

Given : A piece of paper is in the form of a sector, making an angle θ.

The paper is rolled to form a right circular cone of radius 5 cm and height 12 cm

To Find : value of θ   in radians

Solution :

arc Length of given sector = circumference of base of cone

=> arc Length of given sector =  2π(5) = 10π

Radius of sector = Slant height of cone

=> Radius of sector = √5² + 12²  = 13

arc Length of given sector  = ( θ  /2π) * 2π r

=>  10π = ( θ  /2π) * 2π 13

=> 10π =  13θ

=> θ =  10π/ 13

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