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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]
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Solution 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
Solution 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.
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