A metallic spherical shell of internal and external diameters 4 cm and 8 cm, respectively is melted and recast into the form a cone of base diameter 8 cm. The height of the cone is

Options

A. 12 cm
B. 14 cm
C. 15 cm
D. 18 cm

Answer

B

Explanation

Given, internal diameter of spherical shell = 4 cm
and external diameter of shell = 8 cm
`therefore` Internal radius of spherical shell, `r_(1) = (4)/(2) cm = 2 cm" "[therefore "diamter" = 2 xx radius]`
and external radius of shell ,`r_(2) = (8)/(2) = 4cm" "[therefore "diameter" = 2 xx "radius"]`

Now, Volume of the spherical shell `= (4)/(3) pi [r_(2)^(3) - r_(1)^(3)]`
[`therefore` volume of the spherical shell `= (4)/(3) pi { ("external radius")^(3)- ("internal radius ")^(3)`}]
`= (4)/(3) pi(4^(3) - 2^(3))`
`= (4)/(3) pi (64 - 8)`
`= (224)/(4) pi cm^(3)`
Let height of the cone = h cm
Diameter of the base of cone = 8 cm
`therefore` Radius of the base of cone `= (8)/(2) = 4 cm" "[therefore "diameter " = 2 xx "radius"]`
According to the question ,
Volume of cone = Volume of spherical shell
`rArr" " (1)/(3) pi (4)^(2)h= (22)/(4) = 14 cm " "[therefore " of cone" = (1)/(3) xx pi xx ("radius")^(2) xx ("height")]`
Hence, the height of the cone is 14 cm.