Question

The total area of a solid metallic sphere is 1256 cm². It is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate: the number of cones recasted [π = 3.14]

Answer 1

∴ r = 10

Volume of sphere=`4/3pir^3`

`= 4/3 xx 22/7 xx 10 xx 10 xx 10`

`= 88000/21 "cm"^3` 

volume of right circular cone = 

`1/3pir^2h`

`= 1/3 xx 22/7 xx (2.5)^2 xx 8`

`= 1100/21 "cm"^3` 

Number of cones 

`= 88000/21 ÷ 1100/21` 

`= 88000/21 xx 21/1100` 

= 80

Answer 2

The total surface area (TSA) of a sphere is given by:

TSA = 4πr²

Given: TSA = 1256 cm² and π = 3.14

1256 = 4 × 3.14 × r²

`r^2 = 1256/(4 xx 3.14)`

= `1256/12.56`

= 100

`r = sqrt100`

r = 10 cm

The radius of the sphere is 10 cm.

The volume of a sphere is given by:

`"V" = 4/3πr^3`

Substitute r = 10 cm r = 10 and π = 3.14

V = `4/3 xx 3.14 xx (10^3)`

V = `4/3 xx 3.14 xx 1000`

V = `(4 xx 3.14 xx 1000)/3`

= `12560/3`

= 4186.67 cm³

The volume of the sphere is 4186.67 cm³

The volume of a cone is given by:

V = `1/3 πr^2h`

Given:

r = 2.5 cm and h = 8 cm

V = `1/3 xx 3.14 xx (2.5)^2 xx 8`

V = `1/3 xx 3.14 xx 6.25 xx 8`

= `1/3 xx 3.14 xx 50`

V = `157/3`

= 52.33 cm³

The volume of one cone is 52.33 cm³

The number of cones is given by:

`"Number of cones" = " Volume of sphere"/"Volume of one cone"`

Number of cones = `4186.67/52.33`

= 80

The number of cones recast is 80.