We are given that quadrilaterals $A$, $B$, and $C$ are similar. This means that their corresponding side lengths are proportional.

Step-by-step solution:

1. Identify the known dimensions:

• Quadrilateral $A$:
  • Height = 3 m
  • Length = 5 m
• Quadrilateral $B$:
  • Height = 32 m
  • Length = 20 m
• Quadrilateral $C$:
  • Height = 36 m
  • Length = $t$ (unknown).

2. Set up the proportion:

Since the quadrilaterals are similar, we can set up a proportion using the corresponding sides.

We use the ratio of heights to find the unknown length $t$. The height of quadrilateral $C$ is 36 m, and the height of quadrilateral $B$ is 32 m.

$\frac{t}{20} = \frac{36}{32}$

3. Solve for $t$:

Simplify the fraction on the right-hand side:

$\frac{36}{32} = \frac{9}{8}$

Now, solve for $t$ by multiplying both sides of the equation by 20:

$t = 20 \times \frac{9}{8}$

$t = \frac{180}{8} = 22.5$

Final answer:

The unknown length $t$ is 22.5 m.

Would you like more details or have any questions? Here are some related questions:

1. How do you find the scale factor between two similar shapes?
2. What are some properties of similar quadrilaterals?
3. Can this method be applied to 3D shapes as well, and how?
4. What other relationships exist between similar polygons besides their side lengths?
5. How can proportions help solve real-world problems involving similar shapes?

Tip: When working with similar shapes, always identify corresponding sides and use proportions to find unknown values.