1. **State the problem:** We have a right triangle formed by the observer's position $O$, the base of the tower $R$, and the top of the tower $T$. Given: $OR=84$ m, initial angle of elevation $52^\circ$, and after moving back, angle of elevation $49^\circ$. We need to find: (a) height $RT$ of the tower to 3 significant figures. (b) distance observer moved backwards along line $OR$ to 2 decimal places. 2. **Diagram and notation:** - The tower height $RT$ is vertical. - $OR$ is horizontal distance from observer to tower base. - Initial right triangle $ORT$ with angle $52^\circ$ at $O$. 3. **(a) Calculate height $RT$:** Since $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$, $$\tan(52^\circ) = \frac{RT}{84}$$ So, $$RT = 84 \times \tan(52^\circ)$$ Calculate using a calculator: $$RT = 84 \times 1.279942 = 107.515$$ Rounded to 3 s.f.: $$RT = 108 \text{ m}$$ 4. **(b) Distance observer moved back:** Let the new position be $O'$ at distance $x$ meters further from $R$. So new distance $O'R = 84 + x$. Using the new angle $49^\circ$, $$\tan(49^\circ) = \frac{RT}{84 + x}$$ We know $RT=108$ (from part a), so $$\tan(49^\circ) = \frac{108}{84 + x}$$ Rearranged: $$84 + x = \frac{108}{\tan(49^\circ)}$$ Calculate denominator: $$\tan(49^\circ) = 1.150368$$ So, $$84 + x = \frac{108}{1.150368} = 93.913$$ Therefore, $$x = 93.913 - 84 = 9.913$$ Rounded to 2 decimal places: $$x = 9.91 \text{ m}$$ **Final answers:** (a) height of tower $= 108$ m (b) observer moved back $= 9.91$ m