The problem involves two congruent triangles $\triangle ABC \cong \triangle EDC$, where $C$ is the midpoint of $\overline{AE}$ and $\overline{DB}$. Here’s the summary and solution:

Given:

1. Boat A is $9$ miles from Boat B.
   $AB = 9 \, \text{miles}.$
2. Boat A is $30$ miles from object $E$.
   $AE = 30 \, \text{miles}.$
3. $C$ is the midpoint of $\overline{AE}$ and $\overline{DB}$.
   $AC = CE = \frac{30}{2} = 15 \, \text{miles}.$

To find:

The length of $\overline{DB}$, rounded to the nearest mile.

Solution:

Since $\triangle ABC \cong \triangle EDC$, we know the following properties:

• Corresponding sides of congruent triangles are equal.
• $AB = ED = 9 \, \text{miles}$.

Also, $C$ is the midpoint of $\overline{DB}$, so: $DC = CB = \frac{\overline{DB}}{2}.$

Using the Pythagorean theorem in $\triangle ABC$ (or equivalently $\triangle EDC$), the hypotenuse $AB = 9$ forms one leg of the triangle, while $AC = 15$ forms another leg. Solving for $BC$: $BC = \sqrt{AC^2 - AB^2}.$ Substituting the values: $BC = \sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144} = 12 \, \text{miles}.$

Since $\overline{DB}$ consists of $2 \times BC$: $DB = 2 \cdot 12 = 24 \, \text{miles}.$

Final Answer:

The length of $\overline{DB}$ is 24 miles.

Do you have any questions or want more details?
Here are 5 related questions to expand your understanding:

1. What is the significance of congruent triangles in this problem?
2. How does the Pythagorean theorem apply here?
3. Why is $\overline{DB}$ calculated as $2 \cdot BC$?
4. How would the solution change if $AB$ or $AE$ were different?
5. Can you apply similar methods to calculate other distances in geometry?

Tip: When solving problems involving congruent triangles, use the congruency to equate corresponding sides or angles, simplifying the calculations.