1. **State the problem:** We have two similar trapezoidal prisms with corresponding linear dimensions. We need to find the unknown depth $x$ of the smaller prism. 2. **Identify corresponding dimensions:** The smaller prism has depth $x$ mm, height 5 mm, base length 6 mm, and depth 3 mm. The larger prism has depth 18 mm, height 11 mm, and base length 24 mm. 3. **Use similarity ratio:** Since the prisms are similar, the ratios of corresponding linear dimensions are equal: $$\frac{x}{18} = \frac{3}{11} = \frac{6}{24}$$ 4. **Check ratios for consistency:** Simplify $\frac{6}{24} = \frac{1}{4}$ and $\frac{3}{11}$ is approximately 0.2727, so the correct ratio to use is $\frac{3}{11}$ because depth 3 mm corresponds to depth 18 mm. 5. **Set up proportion for $x$:** $$\frac{x}{18} = \frac{3}{11}$$ 6. **Solve for $x$:** Multiply both sides by 18: $$x = 18 \times \frac{3}{11}$$ 7. **Calculate $x$:** $$x = \frac{18 \times 3}{11} = \frac{54}{11}$$ 8. **Simplify fraction:** $\frac{54}{11}$ is already in simplest form. **Final answer:** $$x = \frac{54}{11}$$