To solve this problem, we need to determine the length, breadth, and height of the cuboid box, and then find the ratio of the smallest possible volume to the largest possible volume.

Step 1: Understand the given information

Let the dimensions of the cuboid be $l$ (length), $b$ (breadth), and $h$ (height). We are given the areas of two adjacent faces:

• Area 1 = $lb = 36c \times m^2$
• Area 2 = $bh = 45c \times m^2$

Since $c \times m^2$ is a common factor, let's consider the equations without it for simplicity:

• $lb = 36$
• $bh = 45$

Step 2: Factorize the given areas

Factorize the numbers 36 and 45 to find the possible dimensions $l, b,$ and $h$.

• $36 = 2^2 \times 3^2$
• $45 = 3^2 \times 5$

Now, let’s identify pairs of factors that could represent the possible lengths, breadths, and heights:

• Possible values of $l$ and $b$ (from $lb = 36$) are $(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)$.
• Possible values of $b$ and $h$ (from $bh = 45$) are $(1, 45), (3, 15), (5, 9)$.

Step 3: Match the common dimension $b$

We need to find a common value of $b$ from both equations.

• If $b = 9$, then $l = 4$ (since $4 \times 9 = 36$) and $h = 5$ (since $9 \times 5 = 45$).
• If $b = 3$, then $l = 12$ (since $12 \times 3 = 36$) and $h = 15$ (since $3 \times 15 = 45$).

Step 4: Calculate volumes

The volume $V$ of a cuboid is given by $V = l \times b \times h$.

• For $(l, b, h) = (4, 9, 5)$, the volume is $V_1 = 4 \times 9 \times 5 = 180 \, \text{cm}^3$.
• For $(l, b, h) = (12, 3, 15)$, the volume is $V_2 = 12 \times 3 \times 15 = 540 \, \text{cm}^3$.

Step 5: Find the ratio of the smallest volume to the largest volume

The smallest volume is $180 \, \text{cm}^3$ and the largest volume is $540 \, \text{cm}^3$. The ratio is: $\text{Ratio} = \frac{180}{540} = \frac{1}{3}$ So, the correct answer is $\frac{1}{3}$, which corresponds to option (c) 1:3.

Would you like more details on any part of this solution? Here are five related questions to further your understanding:

1. How would the solution change if the areas given were different?
2. Can you derive the same ratio using different factorization techniques?
3. How do you determine which factorization pairs are valid for the dimensions of a cuboid?
4. What if one of the faces' areas was a non-integer?
5. How would the problem change if the box was a cube instead of a cuboid?

Tip: Always factorize the given values and try to match the common dimension when solving problems involving the dimensions of cuboids or other geometric shapes.