Expert: Ahmed Salami - 10/1/2010

Question
QUESTION: Suppose that the sum of the surface area of a sphere and a
cube is a constant.
(a) Show that the sum of their volumes is smallest when the
diameter of the sphere is equal to the length of an edge of
the cube.
(b) When will the sum of their volumes be greatest?

ANSWER: Hi Jane,
The surface area of a sphere with radius r,
S(r) = 4#r^2       where # represents pi
The surface area of a cube with length l,
S(l) = 6l^2
The surface area sum is
S = S(r) + S(l)
= 4#r^2 + 6l^2
from which
6l^2 = S - 4#r^2
l^2 = (S - 4#r^2)/6
l = [(S - 4#r^2)/6]^(1/2)
The volume of a sphere with radius r,
V(r) = (4/3)#r^3
The volume of a cube with length l,
V(l) = l^3
The volume sum is
V = V(r) + V(l)
= (4/3)#r^3 + l^3
= (4/3)#r^3 + [(S - 4#r^2)/6]^(1/2)^3
= (4/3)#r^3 + [(S - 4#r^2)/6]^(3/2)
V is minimum when dV/dr = 0
dV/dr = 3(4/3)#r^2  +  (3/2)[(S - 4#r^2)/6]^(1/2) .(-8#r/6)
    = 4#r^2  -  2#r[(S - 4#r^2)/6]^(1/2)
at dV/dr = 0
4#r^2  -  2#r[(S - 4#r^2)/6]^(1/2) = 0
4#r^2  =  2#r[(S - 4#r^2)/6]^(1/2)
dividing by 2#r
2r = [(S - 4#r^2)/6]^(1/2)
2r = l
but 2r is the diameter of the sphere and we've shown that it should be equal to the length of an edge of the cube for minimum volume.

Note that r = 0 is also a solution of dV/dr = 0 and so that is another turning point.

You can always get back to me.

Regards

---------- FOLLOW-UP ----------

QUESTION: Thank you so much!!! This helped a lot.
But in question(b) it asks when will the sum of their volumes
be greatest. Do you think you could answer that? I'm a little
stuck on that one.

Answer
Hi Jane,
The sum of their volumes is greatest at the only other turning point. The equation dV/dr = 0 yielded two results; r = 0 and 2r = l which give the maximum and minimum values of the total volume V.
Of course in this particular example we didnt need to prove which is maximum or minimum but we could always do that by taking the second differential dēV/drē and checking its sign at the points.
Anyway, the sum of their volumes is greatest when r = 0.

Regards