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The two bases of a right conical frustum have radii r and R  The frustum has \(\frac{2}{3}\) the volume of a cylinder with the same height as the frustum and a radius of R  Compute r/R.

 

 May 2, 2022
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Volume of the cylinder =  pi * R^2 * h

 

So  the frustum's   volume =  (2/3)* pi * R^2 * h        (1)

 

Volume of a frustum =    (1/3) * pi * h *  [  R + r + Rr ]      (2)

 

Set  (1)   = (2)

 

(2/3)pi*R^2*h =   (1/3)pi * h * [ R + r + Rr ]         simplify

 

2R^2  =  R + r + Rr

 

2R^2 - R  =   r + Rr

 

R (2R - 1 ) =  r (1 + R)

 

R ( 2R - 1 ) / (1 + R)  = r

 

r / R =      R(2R - 1)  /   [ (1 + R) * R ] =

 

(2R - 1) / (R + 1)

 

 

cool cool cool

 May 2, 2022

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