1. **State the problem:** Ravish measures the angle of elevation to the roof of a building and to the center of a logo on the building from point P, which is 24 m from the building's base. The angle of elevation to the roof is 45° and to the logo center is 30°. 2. **Formula and explanation:** The angle of elevation $A$ to the top of the building is given by the angle between the horizontal ground and the line of sight to the roof. We use the tangent function in right triangles: $$\tan(\theta) = \frac{\text{height}}{\text{distance}}$$ where $\theta$ is the angle of elevation, height is the vertical height from ground to the point observed, and distance is the horizontal distance from point P to the building base. 3. **Calculate height of the roof:** Given $\theta = 45^\circ$ and distance $= 24$ m, $$\tan(45^\circ) = \frac{h_{roof}}{24}$$ Since $\tan(45^\circ) = 1$, $$1 = \frac{h_{roof}}{24} \implies h_{roof} = 24 \text{ m}$$ 4. **Calculate height of the logo center:** Given $\theta = 30^\circ$ and distance $= 24$ m, $$\tan(30^\circ) = \frac{h_{logo}}{24}$$ Since $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, $$\frac{1}{\sqrt{3}} = \frac{h_{logo}}{24} \implies h_{logo} = \frac{24}{\sqrt{3}} = 24 \times \frac{\sqrt{3}}{3} = 8\sqrt{3} \text{ m}$$ 5. **Summary:** The angle of elevation $A$ to the top of the building is $45^\circ$ as given, and the heights calculated confirm the measurements. **Final answer:** The angle of elevation $A$ of the top of the building is $\boxed{45^\circ}$.