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The volume of the apple juice in a glass whose shape is an inverted frustum of a cone is 1176π mm³. The depth of the juice is 18mm.Find the radius of the lower base if the product of its radii is 60 mm². 

 Jan 23, 2021
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Volume  frustum  =     (1/3)  pi  * height  (  r^2  +Rr +  R^2)

 

height  = 18

Rr = 60

Note that  2Rr =   120  

 

We have  that

 

1176pi  =   (pi/3)   ( r^2  + Rr  +  R^2)

 

3528  =   r^2 + R^2   +  60

 

3468  =   r^2 + R^2

 

Note that

 

r^2 + 2Rr  + R^2 =  3468  + 2Rr             factor the  left  side

 

(r + R)^2   =   3468  + 120

 

(r + R)^2  =  3588       take  the positive  root

 

r + R   =   sqrt (3588)  

 

R  =  sqrt ( 3588)  -  r

 

The product of the  roots =  60

 

So

 

rR  = 60

 

r ( sqrt (3588)  - r)  = 60

 

-r^2  + sqrt (3588)r   - 60    =  0        mu;tply through  by  -1

 

r^2  - sqrt (3588)r + 60    =  0

 

A graph is the easiest way to solve this  : https://www.desmos.com/calculator/98tdpymtxf

 

Solving  this for r   gives that   r  ≈ 1.019   or  r ≈  58.881

 

Take the  smaller  value     r ≈ 1.019  mm

 

 

cool cool cool

 Jan 23, 2021

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