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Question
A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of two of the balls are 1.5 cm and 2 cm. Find the diameter of the third ball.
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Solution
The radius of spherical ball = 3 cm.
Volume of spherical ball = `4/3` πr3
= `4/3` π x 3 x 3 x 3
= 36 π cm3
∵ Volume of spherical ball = Total volume of three small spherical ball
∵ The radii of the ball are 1.5 cm and 2 cm.
∴ Let the radius of third ball = r
∴ Volume of spherical ball = Total volume of three small spherical balls.
36π = `4/3π (3/2)^3 + 4/3 π (2)^3 + 4/3 π r^3`
36π = `4/3π xx 27/8 + 4/3 π xx 8 + 4/3 π r^3`
36π = `4/3π ( 27/8 + 8 + r^3 )`
`(36π xx 3)/(4π) = 27/8 + 8 + r^3`
27 = `(27 + 64)/8 + r^3`
27 = `91/8 + r^3`
`27 - 91/8 = r^3`
`(216 - 91)/8 = r^3`
`125/8 = r^3`
r = `root(3)(125/8)`
r = `5/2` cm.
The diameter of the third ball = 2r = 2 x `5/2` = 5 cm.
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