1. **State the problem:** We have a right triangle formed by points O (observer), R (base of tower), and T (top of tower). Given: - |OR| = 84m (horizontal distance from observer to base of tower) - Angle of elevation from O to T initially is 52° - After moving backward, angle of elevation is 49° We need to find: a. The height of the tower (TR) to 3 significant figures. b. How far the observer moved backwards along the line OR to 2 decimal places. 2. **Find the height of the tower (TR) at initial position:** Use trigonometry in right triangle OTR. $$\tan(52^\circ) = \frac{\text{height } TR}{\text{base } OR} = \frac{TR}{84}$$ Solve for $TR$: $$TR = 84 \times \tan(52^\circ)$$ Calculate: $$TR \approx 84 \times 1.2799 = 107.5116$$ Rounded to 3 significant figures: $$TR \approx 108\text{ meters}$$ 3. **Find the new distance of the observer from the tower:** Let new distance $OR' = x$ meters. Again, using tangent: $$\tan(49^\circ) = \frac{TR}{x}$$ Solve for $x$: $$x = \frac{TR}{\tan(49^\circ)}$$ Calculate $\tan(49^\circ)$: $$\tan(49^\circ) \approx 1.1504$$ Substitute $TR = 107.5116$: $$x = \frac{107.5116}{1.1504} \approx 93.4389$$ 4. **Calculate how far the observer moved backwards:** Original distance was 84 m. New distance is approx 93.44 m. Distance moved backwards = $93.4389 - 84 = 9.4389$ m Rounded to 2 decimal places: $$9.44 \text{ meters}$$ **Final answers:** a. Height of tower $TR \approx 108$ m (3 s.f.) b. Observer moved backwards $9.44$ m (2 d.p.)