9. From a point on the floor, the angle of elevation to the top of a doorway is \( 43^{\circ} \). The angle of elevation to the ceiling directly above the doorway is \( 56^{\circ} \). If the ceiling is 10 feet above the floor, how high is the doorway? How far in front of the doorway were the angles of elevation measured?

To solve this problem, we can use some trigonometry! Let's denote the height of the doorway as \( h_d \) and the horizontal distance from the point on the floor to the base of the doorway as \( d \). 1. **Using the angle of elevation to the ceiling:** We have the angle of elevation to the ceiling \( 56^{\circ} \). \[ \tan(56^{\circ}) = \frac{\text{opposite}}{\text{adjacent}} = \frac{10}{d} \] Rearranging it gives us: \[ d = \frac{10}{\tan(56^{\circ})} \] 2. **Using the angle of elevation to the top of the doorway:** The angle of elevation to the top of the doorway is \( 43^{\circ} \). \[ \tan(43^{\circ}) = \frac{h_d}{d} \] Substitute \( d \) from our previous equation: \[ h_d = d \cdot \tan(43^{\circ}) = \frac{10 \cdot \tan(43^{\circ})}{\tan(56^{\circ})} \] Now, you can plug in the values of \( \tan(43^{\circ}) \) and \( \tan(56^{\circ}) \) from a calculator to find \( d \) and \( h_d \). Calculating these out gives: \[ d \approx \frac{10}{1.4826} \approx 6.73 \text{ feet} \] Then for \( h_d \): \[ h_d \approx \frac{10 \cdot 0.9325}{1.4826} \approx 6.29 \text{ feet} \] So, the height of the doorway is approximately **6.29 feet**, and the point from which the angles were measured is about **6.73 feet** in front of the doorway.