Volumes question (It's hard) (1 Viewer)

[cutemouse]

A circular hole is drilled through the centre of a sphere of radius 1cm so that the volume of the sphere cut away is half of the volume of the sphere. Find the radius of the drill bit used.

 

[gurmies]

Welcome back 3unitz <3

 

[Aerath]

Alternative method methinks.

 

[shaon0]

View attachment 19000

volume of cut-out = volume cylinder part + 2*dome part

(2/3)*pi = pi*r^2*[2*sqrt(1 - r^2)] + 2V

(2/3)*pi = pi*r^2*[2*sqrt(1 - r^2)] + 2*pi*[(2/3) - (1 - r^2)^(1/2) + (1/3)*(1 - r^2)^(3/2)]

let u = sqrt(1 - r^2)

(2/3)*pi = pi*(1 - u^2)*2*u + 2*pi*[(2/3) - u + (1/3)*u^3]

1/3 = u - u^3 + (2/3) - u + (1/3)*u^3

(2/3)*u^3 = 1/3

u = (1/2)^(1/3)

sqrt(1 - r^2) = (1/2)^(1/3)

1 - r^2 = (1/2)^(2/3)

r = sqrt[1 - (1/2)^(2/3)]

r ~ 0.608 cm

Finally, the genius is back

 

[cutemouse]

Any idea on where this question came from? (eg. CSSA or those external papers?)

 

[Aerath]

I've seen this question in one of my school's trials (SBHS). Though, I'm fairly sure it's ripped from somewhere else.