The dimensions of a rectangular box are integers greater than 1. If the area of one side of this box is 12 and the area of another side 15, what is the volume of the box?

Answer

To find the volume of the rectangular box given the areas of two sides, we can follow these steps:

1. **Identify the Areas**: We know that the area of one side (base and height) is 12, and the area of another side (length and height) is 15. Let's denote:
- \( b \) = base
- \( h \) = height
- \( l \) = length

From the problem, we have:
\[ b \cdot h = 12 \quad (1) \]
\[ l \cdot h = 15 \quad (2) \]

2. **Express \( b \) and \( l \) in terms of \( h \)**: From equation (1), we can express \( b \) as:
\[ b = \frac{12}{h} \quad (3) \]
From equation (2), we can express \( l \) as:
\[ l = \frac{15}{h} \quad (4) \]

3. **Find the ratio of \( b \) and \( l \)**: To find the ratio of \( b \) to \( l \), we can substitute equations (3) and (4):
\[ \frac{b}{l} = \frac{\frac{12}{h}}{\frac{15}{h}} = \frac{12}{15} = \frac{4}{5} \]

4. **Set \( b \) and \( l \) in terms of a common variable**: Let’s assume \( b = 4k \) and \( l = 5k \) for some integer \( k \).

5. **Substitute back to find \( h \)**: Using equation (3):
\[ 4k \cdot h = 12 \implies h = \frac{12}{4k} = \frac{3}{k} \quad (5) \]
Using equation (4):
\[ 5k \cdot h = 15 \implies h = \frac{15}{5k} = \frac{3}{k} \quad (6) \]
Both equations (5) and (6) confirm that \( h = \frac{3}{k} \).

6. **Determine possible values of \( k \)**: Since \( h \) must be an integer greater than 1, \( k \) can only be 1 (as \( k \) must also be greater than 1). Thus:
\[ k = 1 \implies h = 3 \]
Substituting \( k = 1 \) into \( b \) and \( l \):
\[ b = 4 \times 1 = 4 \]
\[ l = 5 \times 1 = 5 \]

7. **Calculate the Volume**: The volume \( V \) of the box is given by:
\[ V = b \cdot l \cdot h = 4 \cdot 5 \cdot 3 \]
\[ V = 60 \]

Thus, the volume of the box is \( \boxed{60} \).