The problem involves two similar rectangles $P$ and $Q$, and we are asked to find the value of $x$, given the side lengths of the rectangles.

Step-by-step solution:

1. Similar rectangles property: Since the rectangles are similar, the ratio of their corresponding sides must be equal.

   For rectangle $P$, the sides are:

   • Width: $18 \, \text{mm}$
   • Height: $x \, \text{mm}$

   For rectangle $Q$, the sides are:

   • Width: $5x \, \text{mm}$
   • Height: $10 \, \text{mm}$

   The width-to-height ratio of $P$ equals the width-to-height ratio of $Q$: $\frac{18}{x} = \frac{5x}{10}.$

2. Simplify the ratio: The equation can be rewritten as: $\frac{18}{x} = \frac{5x}{10}.$ Cross-multiply: $18 \cdot 10 = x \cdot 5x.$

   Simplify: $180 = 5x^2.$

3. Solve for $x^2$: Divide both sides by 5: $x^2 = \frac{180}{5}.$ $x^2 = 36.$

4. Solve for $x$: Take the square root of both sides: $x = \sqrt{36}.$ $x = 6.$

Final Answer:

$x = 6 \, \text{mm}.$

Let me know if you'd like additional details!

Related questions:

1. How do you calculate the side lengths of similar shapes in general?
2. What happens if the ratio of corresponding sides is not equal in two rectangles?
3. Could this problem be solved using proportions instead of cross-multiplication?
4. What is the geometric meaning of the term "similar" in two-dimensional shapes?
5. How would the answer change if one of the measurements was given in centimeters instead of millimeters?

Tip:

Always double-check the ratios of corresponding sides when dealing with similar shapes to ensure accuracy.