Chapter 12, Problem 1P

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

Each edge of a cubical box has length 1 m. The box contains nine spherical balls with the same radius *r*. The center of one ball is at the center of the cube and it touches the other eight balls. Each of the other eight balls touches three sides of the box. Thus the balls are tightly packed in the box (see the figure). Find *r*. (If you have trouble with this problem, read about the problem-solving strategy entitled *Use Analogy* on page 71.)

To determine

**To find**: The radius of spherical ball.

Explanation

**Given data:**

The length of cube as 1 m.

The square and circle are analogous to cube and sphere in two-dimensional plane respectively.

Re-draw the analogous cross section diagram across diagonal of Figure for problem 1 in two-dimensional plane as shown in Figure 1.

Write the expression for diagonal of cube

Here,

*a* is length of square.

Substitute 1 m for *a* in equation (1),

Write the expression for diagonal from Figure 1.

Here,

*r* is radius of sphere, and

*x* is diagonal of small square.

Substitute *x* for *d* and *r* for *a* in equation (1),

Substitute
*x* in equation (2),

Multiply and divided by term

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