1. **Problem statement:** Leah and Malia stand on opposite sides of a church spire. Leah is 120 m from the spire with an angle of elevation of 18° to the top. Malia's angle of elevation to the top is 32°. We need to find Malia's distance $x$ from the base of the spire. 2. **Understanding the problem:** The spire, Leah, and Malia form a triangle with the spire at the apex. The angles of elevation correspond to angles at Leah's and Malia's positions looking up to the spire's top. 3. **Key formula:** Using the tangent function for right triangles: $$\tan(\theta) = \frac{\text{height of spire}}{\text{distance from spire base}}$$ 4. **Set variables:** Let $h$ be the height of the spire. From Leah's position: $$\tan(18^\circ) = \frac{h}{120} \implies h = 120 \times \tan(18^\circ)$$ From Malia's position: $$\tan(32^\circ) = \frac{h}{x} \implies x = \frac{h}{\tan(32^\circ)}$$ 5. **Calculate $h$:** $$h = 120 \times \tan(18^\circ)$$ Using $\tan(18^\circ) \approx 0.3249$: $$h \approx 120 \times 0.3249 = 38.988$$ 6. **Calculate $x$:** $$x = \frac{38.988}{\tan(32^\circ)}$$ Using $\tan(32^\circ) \approx 0.6249$: $$x \approx \frac{38.988}{0.6249} = 62.38$$ 7. **Answer:** Malia is approximately 62.38 meters from the base of the spire.