SOLUTION: Three spheres of radii 4,5,and 6 inches are placed in a cylinder of diameter 15 inches. Find the minimum volume of water needed to completely submerge the three spheres.

Algebra ->  Bodies-in-space -> SOLUTION: Three spheres of radii 4,5,and 6 inches are placed in a cylinder of diameter 15 inches. Find the minimum volume of water needed to completely submerge the three spheres.      Log On


   

Question 1025615: Three spheres of radii 4,5,and 6 inches are placed in a cylinder of diameter 15 inches. Find the minimum volume of water needed to completely submerge the three spheres.
Found 2 solutions by Fombitz, Edwin McCravy:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
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From the illustration,
6%2BA%2B5=15
A=4
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5%2BC%2B4=15
C=6
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Let's define alpha as the angle whose tangent is equal to,
tan%28alpha%29=B%2FA%29
You also know that,
cos%28alpha%29=A%2F11
cos%28alpha%29=4%2F11
alpha=68.7
So then,
B=11sin%2868.7%29
B=10.25
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Let's define beta as the angle whose tangent is equal to,
tan%28beta%29=D%2FC
You also know,
cos%28beta%29=C%2F9
cos%28beta%29=6%2F9
beta=48.2
So then similarly,
D=9sin%2848.2%29
D=6.71
So know we can calculate the depth (as shown by the red line).
D=6%2BB%2BD%2B4
D=6%2B10.25%2B6.71%2B4
D=26.96
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So now the entire volume is made up of water and spheres and is equal to,
V=pi%2A%2815%2F2%29%5E2%2A26.96
V=4764.22in%5E3
V%5Bs1%5D%2BV%5Bs2%5D%2BV%5Bs3%5D%2BV%5Bw%5D=4764.22
%284%2F3%29pi%28r%5B1%5D%5E3%2Br%5B2%5D%5E3%2Br%5B3%5D%5E3%29%2BV%5Bw%5D=4764.22
%284%2F3%29pi%284%5E3%2B5%5E3%2B6%5E3%29%2BV%5Bw%5D=4764.22
1696.46%2BV%5Bw%5D=4764.22
highlight_green%28V%5Bw%5D=3067.8%29in%5E3

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

The other tutor did not get the minimum volume of water. When we
reverse the upper two spheres the spheres do not stick up quite
as high in the cylinder.  On the left below is his solution drawn to
scale and on the right is the case when the upper two spheres
are reversed, also drawn to scale. You can see that not quite as
much water will be needed to completely submerge the three spheres
when the smallest sphere is placed between the two larger ones.

I will not solve it for you.  All you need do is follow his procedure
above to find the minimum volume of water, as it is done exactly the
same way.  Only the numbers will be different.

  

Edwin