What is combined volume of the slices?

Question

A spherical water mellon is dissected by three parallel
cuts into two spherical segments, and two slices,
each slice is h = 2 cm thick. The areas of the bases
of the slices are 210, 305, 370 cm^2.

What is combined volume of the slices?

305 is the middle common area.

Answer 1

There does not seem to be a solution for the data given. Call the sphere radius R; the radius of the three slices r1, r2, and r3; the position of the central slice (relative to the center) z; and the slice thickness h. The relation for the central slice is:

R^2 = r^2 + z^2

The slice below that is: R^2 = r^2 + (z - h)^2
The slice above is: R^2 = r^2 + (z + h)^2

The regular inknowns are the sphere radius (R) and the location of the central slice (z). In addition, we do not know which slice is which. I've used r in teh equations above. These must be r1, r2, and r3 but we don't know which is which. There are 6 possible ways to put the known radii in the equations. We are given the areas A and the radii are:

r = 4A/pi

Call the radii ra, tb and rc in the equations, giving:

R^2 = ra^2 + z^2
R^2 = rb^2 + (z - h)^2
R^2 = rc^2 + (z + h)^2

Combin these by subtracting the first from each the second and third:

0 = (rb^2 + (z - h)^2) - (ra^2 + z^2)
0 = rb^2 - ra^2 - 2zh + h^2

z = (rb^2 - ra^2 + h^2)/(2h)

0 = (rc^2 + (z + h)^2) - (ra^2 + z^2)
0 = rc^2 - ra^2 + 2zh + h^2

z = -(rc^2 - ra^2 + h^2)/(2h)

For a solution to exist, there must be a matching of r1, r2, and r3 to ra, rb, and rc such that the two equations for z match. Equating the z relations:

(rb^2 - ra^2 + h^2)/(2h) = -(rc^2 - ra^2 + h^2)/(2h)
rb^2 - ra^2 + h^2 = -(rc^2 - ra^2 + h^2)
rb^2 - ra^2 + h^2 = -(rc^2 - ra^2 + h^2)
rb^2 + rc^2 - 2ra^2 + 2h^2 = 0

SInce all the radii are of the form r^2 = A/pi and h is an integer no solution can exist. If you recast the problem as:

A1 = 210*pi, A2 = 305*pi, A3 = 370*pi

At least a solution is possible for integer values. It appears that the difference in slice areas is too big compared to the slice thickness. The average difference between slices is only h^2 (4 here) while you have a difference of about 80.

Answer 2

I am not sure why the areas are given. The sum total of the thicknesses is the diameter of the original sphere.
=>d = 8 cm and r = 4 cm
=>V = (4/3)*pi*4^3 = 268.082 cm^3

But the areas provided seem to suggest a larger volume. Are u sure the question is correct?

Answer 3

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