Los encargados de una montaña donde se practica el esquí desean construir un nuevo remonte desde B  hasta C como se muestra en la figura siguiente. Desde A hasta C hay 1150 pies de distancia. El ángulo DAC  mide 30°  y el ángulo DBC mide 24° . ¿Cuál es la distancia desde A  hasta B ? Redondear la respuesta a la décima de pie más cercana.


To solve this problem, we need to determine the distance \( AB \) using the given angles and the distance \( AC = 1150 \) feet.

### Given:
- \( AC = 1150 \) feet
- \( \angle DAC = 30^\circ \)
- \( \angle DBC = 24^\circ \)

### Approach:
This problem involves using the Law of Sines in a non-right triangle \( \triangle ABC \).

#### Step 1: Calculate the Angle \( \angle ABC \)
Since \( \angle DAC \) is given, let's first find \( \angle BAC \).

- \( \angle BAC = 180^\circ - \angle DAC = 180^\circ - 30^\circ = 150^\circ \)

#### Step 2: Apply the Law of Sines
The Law of Sines states:
\[
\frac{AB}{\sin(\angle ABC)} = \frac{AC}{\sin(\angle BAC)}
\]
We need to calculate \( \angle ABC \), which is complementary to \( \angle DBC \) within \( \triangle ABC \):
\[
\angle ABC = 180^\circ - \angle DBC = 180^\circ - 24^\circ = 156^\circ
\]

Now applying the Law of Sines:
\[
\frac{AB}{\sin(156^\circ)} = \frac{1150 \text{ feet}}{\sin(30^\circ)}
\]

#### Step 3: Solve for \( AB \)
\[
AB = \frac{1150 \times \sin(156^\circ)}{\sin(30^\circ)}
\]

Using a calculator:
- \( \sin(156^\circ) \approx 0.5 \)
- \( \sin(30^\circ) = 0.5 \)

\[
AB = \frac{1150 \times 0.5}{0.5} = 1150 \text{ feet}
\]

This calculation shows the distance \( AB \) to be approximately \( 1150 \) feet. However, the scenario might be more complex if a different interpretation of the angles is used. It would be important to confirm the interpretation or if additional steps, like checking angles relative to the horizontal or vertical directions, should be considered.

### Conclusion
The distance from \( A \) to \( B \) is approximately \( 1150 \) feet.

Would you like any additional details or clarification on this solution?

**Next steps:**
1. Revalidate angles \( \angle ABC \) and \( \angle BAC \) to confirm the correct triangle application.
2. Confirm or modify the problem's initial conditions or assumptions.
3. Explore different solution methods, such as using trigonometry for verifying the result.
4. Consider special cases where angle or distance assumptions might change.
5. Review the concepts of the Law of Sines and Cosines in different contexts.

**Tip:** Always ensure all angles in non-right triangles add up to \( 180^\circ \) when applying trigonometric laws.

Learn how to calculate the distance AB in a triangle ABC using trigonometry and the Law of Sines. Given angles DAC = 30° and DBC = 24°, and distance AC = 1150 feet, find the distance AB rounded to the nearest tenth of a foot.

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