1. **State the problem:** We have a lighthouse of height $h$ and two points $A$ and $B$ on the horizontal line from the base of the lighthouse. The horizontal distance from $A$ to the lighthouse base is 1315 feet. The angle of elevation from $A$ to the top of the lighthouse is $12^\circ$, and from $B$ it is $8^\circ$. We need to find the distance from $A$ to $B$. 2. **Identify known values and variables:** - Height of lighthouse: $h$ (unknown) - Distance from $A$ to lighthouse base: $1315$ ft - Distance from $B$ to lighthouse base: $x$ ft (unknown) - Angle of elevation at $A$: $12^\circ$ - Angle of elevation at $B$: $8^\circ$ 3. **Use the tangent function:** The tangent of the angle of elevation equals the opposite side (height $h$) over the adjacent side (horizontal distance). From point $A$: $$\tan(12^\circ) = \frac{h}{1315} \implies h = 1315 \tan(12^\circ)$$ From point $B$: $$\tan(8^\circ) = \frac{h}{x} \implies h = x \tan(8^\circ)$$ 4. **Set the two expressions for $h$ equal:** $$1315 \tan(12^\circ) = x \tan(8^\circ)$$ 5. **Solve for $x$:** $$x = \frac{1315 \tan(12^\circ)}{\tan(8^\circ)}$$ 6. **Calculate $x$ using approximate tangent values:** $$\tan(12^\circ) \approx 0.2126, \quad \tan(8^\circ) \approx 0.1405$$ $$x = \frac{1315 \times 0.2126}{0.1405} \approx \frac{279.4}{0.1405} \approx 1987.9 \text{ feet}$$ 7. **Find the distance from $A$ to $B$:** Since $A$ is 1315 feet from the lighthouse and $B$ is $x \approx 1987.9$ feet from the lighthouse, and $B$ lies between $A$ and the lighthouse, the distance from $A$ to $B$ is: $$|1987.9 - 1315| = 672.9 \approx 673 \text{ feet}$$ **Final answer:** The distance from point $A$ to point $B$ is approximately **673 feet**.