To find the scale of the model in the form $1:n$, we need to consider the relationship between the volumes of similar objects, as the model and the actual plane are geometrically similar.

The formula for the scale factor in terms of volume is:

$\left( \frac{\text{Model Length}}{\text{Actual Length}} \right)^3 = \frac{\text{Volume of Model}}{\text{Volume of Actual}}$

Given:

• Volume of the model plane = $1200 \, \text{cm}^3$
• Volume of the full-size plane = $4050 \, \text{m}^3 = 4,050,000 \, \text{cm}^3$ (converting to cubic centimeters for consistency)

Now, we calculate the scale factor in terms of volume:

$\frac{\text{Volume of Model}}{\text{Volume of Actual}} = \frac{1200}{4050000}$

Simplifying this ratio:

$\frac{1200}{4050000} = \frac{1}{3375}$

Since this ratio is for volumes, we need to take the cube root to find the linear scale factor:

$\sqrt[3]{\frac{1}{3375}} = \frac{1}{15}$

Thus, the scale of the model is $1:15$.

Would you like further details on this solution?

Here are some related questions to consider:

1. How would the calculation change if the volumes were given in different units?
2. What is the linear scale factor if only the lengths, rather than volumes, were given?
3. How does scaling affect other properties, like surface area, in similar objects?
4. What if the model's volume was provided in liters instead of cubic centimeters?
5. How would we proceed if only the surface area ratio were given?

Tip: When dealing with scaling problems, remember that volume scales by the cube of the linear scale factor, while area scales by the square.