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Chapter 10: Problem 145

For a simple cubic array, solve for the volume of an interior sphere (cubic hole) in terms of the radius of a sphere in the array.

Short Answer

Expert verified
The volume of the interior sphere (cubic hole) in terms of the radius of a sphere in the array is \(512r^3(\sqrt{22} - 1)^3\).

Step by step solution

01

Identify the geometric relationship between the spheres in the array and the cubic hole

In a simple cubic array, spheres are arranged in such a way that their centers form the vertices of a cube. Therefore, if the radius of each sphere in the array is r, the distance between the centers of any two adjacent spheres (and also the edge length of the cube) will be 2r. The cubic hole is located in the center of the cube, surrounded by eight spheres from the array (one at each corner of the cube). The sphere centers and the cubic hole's corners form a smaller cube inside the larger cube formed by the vertex centers.
02

Determine the length of a side of the smaller cube

Drawing a straight line from one vertex of the large cube to the diametrically opposite vertex, we form a body diagonal of the cube. This diagonal goes through the centers of two opposite spheres and the cubic hole in the middle. Using Pythagorean theorem, let's denote the length of this diagonal as d: \(d^2 = (8r)^2 + (8r)^2 + (8r)^2\) \(d^2 = 3(8r)^2\) Now, notice that d is also the length of the side of the smaller cube plus the diameter of the two spheres it passes through (i.e. 4r). Let's denote the length of the side of the smaller cube as x: \(d = x + 4r\)
03

Solve for x

Substitute d from Step 2 and set up an equation to solve for x: \((x + 4r)^2 = 3(8r)^2\) Expand and simplify the equation to solve for x: \(x^2 + 8rx + 16r^2 = 192r^2\) \(x^2 + 8rx = 176r^2\) In this situation, x is positive because it represents a length. Solving for x, we get: \(x = 8r(\sqrt{22} - 1)\)
04

Determine the volume of the cubic hole

Now that we have the length of a side of the smaller cube (which represents the cubic hole), we can find its volume by raising x to the third power: Volume of cubic hole = \(x^3\) Substitute the value of x from Step 3: Volume of cubic hole = \((8r(\sqrt{22} - 1))^3\) Volume of cubic hole = \(512r^3(\sqrt{22} - 1)^3\) So, the volume of the interior sphere (cubic hole) in terms of the radius of a sphere in the array is \(512r^3(\sqrt{22} - 1)^3\).

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