# There are 40 marbles of radius #x#cm each & 60 marbles of radius #y#cm each. If #x# & #y# vary such that #x + y = 15#, how do I find the (exact) value of #x# & #y# that will make the sums of the volumes a minimum?

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All the marbles are spherical. I don't know what the writer completely means by "sum of **the** volume", so, sorry if it's a bit ambiguous.

My current approach is to express #y# in terms of #x# and finding the sum of all 100 marbles through substitution, then setting its derivative to #0# . After taking the derivative, everything just becomes messy. I have a feeling that the actual solution involves the manipulation of the #2:3# ratio of the marbles but I don't know how to work it out.

All the marbles are spherical. I don't know what the writer completely means by "sum of **the** volume", so, sorry if it's a bit ambiguous.

My current approach is to express

##### 1 Answer

the values that makes the volume minimum:

#### Explanation:

The volume of the sphere is given by

volume of 40 spherical marble of a radius x is given by :

volume of 60 spherical marble of a radius y is given by :

differentiate the

find critical values by equating

by using quadratic formula

But the values that makes the volume minimum: