Suppose an orange to be a perfect sphere whose diameter is 3in, and suppose that the skin is of uniform thickness 1/8 in, and that the juice of the orange is uniformly distributed throughout its meat. Find the depth of a slice cut from the outside that will contain as much juice as a slice whose bases are parallel section 1/8 in from the center.

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I assume this is what you're describing?:

The section 1/8th inch on either side of the x-axis must be the same as the segment with height h?

Assuming I am interpreting correctly, find the volume of the slice in the center of the orange.

We can find this by using calculus.

The slices have a base which is bounded by x^{2}+y^{2}=9

A(x)=\pi r^{2}

r=\sqrt{9-x^{2}}

{\pi}\int_{-1/8}^{\1/8}(\sqrt{9-x^{2}})^{2}dx

{\pi}\int_{-1/8}^{1/8}(9-x^{2})dx

Integrate to find the volume of the 1/4 inch slice at the center (1/8th inch above and below the center)

Then, to find the volume of the segment, integrate from h to 3 instead of -1/8 to 1/8. Set this result equal to the volume from the first integral above, solve for h.

I just used 3 inches as the radius of the orange. You may have to adjust to take the skin into account.