The internal base of a rectangular box is 15 cm long and `12 (1)/(2)` cm wide and its height is ` 7 (1)/( 2)` cm . The box is filled with cube each of side ` 2(1)/(2)` cm . The number of cubes will be

Options

A. 90
B. 120
C. 45
D. 60

Answer

To find the number of cubes that can fit into the rectangular box, we first need to calculate the volume of the box and the volume of one cube. Then, we will divide the volume of the box by the volume of one cube to get the number of cubes that can fit.

### Step 1: Convert mixed numbers to improper fractions
- The width of the box is \( 12 \frac{1}{2} \) cm, which can be converted to an improper fraction:
\[ 12 \frac{1}{2} = \frac{25}{2} \text{ cm} \]
- The height of the box is \( 7 \frac{1}{2} \) cm, which can also be converted:
\[ 7 \frac{1}{2} = \frac{15}{2} \text{ cm} \]
- The side of the cube is \( 2 \frac{1}{2} \) cm, which can be converted as well:
\[ 2 \frac{1}{2} = \frac{5}{2} \text{ cm} \]

### Step 2: Calculate the volume of the rectangular box
The volume \( V \) of the box can be calculated using the formula:
\[ V = \text{Length} \times \text{Width} \times \text{Height} \]
Substituting the values:
\[ V = 15 \text{ cm} \times \frac{25}{2} \text{ cm} \times \frac{15}{2} \text{ cm} \]

Calculating step-by-step:
1. Calculate the width and height:
\[ \frac{25}{2} \times \frac{15}{2} = \frac{375}{4} \text{ cm}^2 \]
2. Now multiply by the length:
\[ V = 15 \times \frac{375}{4} = \frac{5625}{4} \text{ cm}^3 \]
\[ V = 1406.25 \text{ cm}^3 \]

### Step 3: Calculate the volume of one cube
The volume \( V_c \) of one cube is given by:
\[ V_c = \text{Side}^3 \]
Substituting the side length:
\[ V_c = \left(\frac{5}{2}\right)^3 = \frac{125}{8} \text{ cm}^3 \]

### Step 4: Calculate the number of cubes that can fit into the box
To find the number of cubes \( N \), we divide the volume of the box by the volume of one cube:
\[ N = \frac{V}{V_c} = \frac{1406.25}{\frac{125}{8}} \]
Calculating this:
1. Convert \( 1406.25 \) to a fraction:
\[ 1406.25 = \frac{5625}{4} \]
2. Now perform the division:
\[ N = \frac{\frac{5625}{4}}{\frac{125}{8}} = \frac{5625 \times 8}{125 \times 4} = \frac{45000}{500} = 90 \]

### Final Answer
The number of cubes that can fit in the box is \( \boxed{90} \).