A rectangular box with length 22 inches , width 5 inches , and height 5 inches is to be packed with steel balls of radius 2 inches in such a way that the centers of the balls are collinear. What is the maximum number of balls that can fit into the box, provided that no balls should protrude from the box ?

Options

A. 0
B. 5
C. 6
D. 10

Answer

To determine the maximum number of steel balls that can fit into the rectangular box, we will follow these steps:

### Step 1: Understand the dimensions of the box and the balls
- The dimensions of the box are:
- Length = 22 inches
- Width = 5 inches
- Height = 5 inches
- The radius of each steel ball is 2 inches.

### Step 2: Calculate the diameter of the balls
- The diameter \(D\) of each ball is given by the formula:
\[ D = 2 \times \text{radius} = 2 \times 2 = 4 \text{ inches} \]

### Step 3: Determine how many balls can fit along the length of the box
- Since the balls are placed in a collinear manner (in a single line), we will first see how many balls can fit along the length of the box.
- The length of the box is 22 inches.
- Each ball occupies 4 inches (its diameter).
- The number of balls that can fit along the length is calculated by:
\[ \text{Number of balls along length} = \frac{\text{Length of the box}}{\text{Diameter of one ball}} = \frac{22}{4} = 5.5 \]
- Since we cannot have a fraction of a ball, we take the whole number:
\[ \text{Maximum balls along length} = 5 \]

### Step 4: Determine how many balls can fit along the width and height of the box
- The width and height of the box are both 5 inches.
- Each ball occupies 4 inches (its diameter).
- The number of balls that can fit along the width is:
\[ \text{Number of balls along width} = \frac{\text{Width of the box}}{\text{Diameter of one ball}} = \frac{5}{4} = 1.25 \]
- Again, we take the whole number:
\[ \text{Maximum balls along width} = 1 \]

- The same calculation applies for the height since it is also 5 inches:
\[ \text{Number of balls along height} = \frac{5}{4} = 1.25 \implies \text{Maximum balls along height} = 1 \]

### Step 5: Calculate the total number of balls that can fit in the box
- The total number of balls that can fit in the box is the product of the maximum number of balls that can fit along the length, width, and height:
\[ \text{Total number of balls} = (\text{Balls along length}) \times (\text{Balls along width}) \times (\text{Balls along height}) = 5 \times 1 \times 1 = 5 \]

### Final Answer
The maximum number of steel balls that can fit into the box is **5**.

---