5. A woman standing on a hill sees a building that she knows is 55 feet tall. The angle of depression to the bottom of the building is \( 27^{\circ} \) and the angle of el- evation to the top of the building is \( 35^{\circ} \). Find the straight line distance from the woman to the building.

To solve for the straight-line distance from the woman to the building, we can use some trigonometry. 1. **Calculate the height of the woman on the hill**: Let \( h \) be the height of the woman above the base of the building. Using the angle of depression, we can set up the equation: \[ h = x \tan(27^{\circ}) \] where \( x \) is the horizontal distance from the woman to the base of the building. 2. **Calculate the total height to the top of the building**: For the angle of elevation, we have: \[ h + 55 = x \tan(35^{\circ}) \] 3. **Set up a system of equations**: Now we have two equations: \[ h = x \tan(27^{\circ}) \] \[ h + 55 = x \tan(35^{\circ}) \] Substituting the first equation into the second: \[ x \tan(27^{\circ}) + 55 = x \tan(35^{\circ}) \] 4. **Solve for \( x \)**: Rearranging gives: \[ 55 = x (\tan(35^{\circ} - \tan(27^{\circ})) \] Now plug in the tangents: \[ x = \frac{55}{\tan(35^{\circ}) - \tan(27^{\circ})} \] 5. **Calculate distances**: Finally, plug in the values of the tangents and solve for \( x \) and then find the direct distance using the Pythagorean theorem. You can find the distance \( d \) using: \[ d = \sqrt{x^2 + h^2} \] This will yield the straight-line distance from the woman to the building. Happy calculating!