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.
+128456 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
