Sixteen cylindrical cans each with a radius of 1 unit are placed inside a cardboard box four in a row. If the cans touch the adjacent cans and or the walls of the box, then which of the following could be the interior area of the bottom of the box in square units ?

Options

A. 16
B. 32
C. 64
D. 128

Answer

To solve the problem, we need to find the interior area of the bottom of a cardboard box that holds 16 cylindrical cans, each with a radius of 1 unit. The cans are arranged in a grid of 4 rows and 4 columns.

### Step-by-Step Solution:

1. **Determine the Diameter of Each Can:**
- The radius of each can is given as 1 unit.
- The diameter (D) of each can can be calculated using the formula:
\[ D = 2 \times \text{radius} \]
- Therefore, the diameter of each can is:
\[ D = 2 \times 1 = 2 \text{ units} \]

**Hint:** Remember that the diameter is twice the radius.

2. **Calculate the Total Length for the Cans in a Row:**
- Since there are 4 cans placed in a row, the total length (L) occupied by the cans can be calculated as:
\[ L = \text{number of cans} \times \text{diameter of each can} \]
- Thus, the total length is:
\[ L = 4 \times 2 = 8 \text{ units} \]

**Hint:** Multiply the number of cans by the diameter to find the total length.

3. **Determine the Dimensions of the Box:**
- Since the cans are placed in a square arrangement (4 in a row and 4 in a column), the width (W) of the box will be the same as the length:
\[ W = L = 8 \text{ units} \]

**Hint:** In a square, all sides are equal.

4. **Calculate the Area of the Bottom of the Box:**
- The area (A) of the bottom of the box can be calculated using the formula for the area of a square:
\[ A = \text{length} \times \text{width} \]
- Therefore, the area is:
\[ A = 8 \times 8 = 64 \text{ square units} \]

**Hint:** For a square, the area is found by squaring the length of one side.

5. **Conclusion:**
- The interior area of the bottom of the box is 64 square units.

### Final Answer:
The interior area of the bottom of the box is **64 square units**.