Math Snap

STEP 1

1. We are given a triangle with points $A$, $B$, and $C$.
2. The angle $\angle DAB = 25^\circ$.
3. The angle $\angle ACB = 15^\circ$.
4. The distance $AC = 1000$ feet.
5. We need to find the distance $AB$.

STEP 2

1. Analyze the given information and determine the triangle type.
2. Use the Law of Sines to find the necessary angles and sides.
3. Calculate the distance from $A$ to $B$ using the derived values.

STEP 3

First, identify the triangle and the given angles. We have a triangle $\triangle ACB$ with $\angle ACB = 15^\circ$ and a line $AD$ such that $\angle DAB = 25^\circ$. The distance from $A$ to $C$ is $1000$ feet.

STEP 4

To find the length $AB$, we need to determine the angle $\angle CAB$. Since $\angle DAB = 25^\circ$ and $\angle ACB = 15^\circ$, the angle $\angle CAB$ is:
$$\angle CAB = \angle DAB - \angle ACB = 25^\circ - 15^\circ = 10^\circ$$

STEP 5

Now, use the Law of Sines in $\triangle ACB$ to find $AB$. The Law of Sines states:
$$\frac{AB}{\sin(\angle ACB)} = \frac{AC}{\sin(\angle CAB)}$$ Substitute the known values:
$$\frac{AB}{\sin(15^\circ)} = \frac{1000}{\sin(10^\circ)}$$

STEP 6

Solve for $AB$:
$$AB = \frac{1000 \cdot \sin(15^\circ)}{\sin(10^\circ)}$$

Calculate the value using a calculator:
$$AB \approx \frac{1000 \cdot 0.2588}{0.1736}$$ $$AB \approx \frac{258.8}{0.1736}$$ $$AB \approx 1490.09$$ The distance from $A$ to $B$ is approximately $\boxed{1490.09}$ feet.