1. **Problem statement:** Bob and Vicky stand 180 m apart with a TV tower between them. The angle of elevation from Bob to the top of the tower is 44° and from Vicky is 53°. We need to find the height of the tower. 2. **Setup:** Let the height of the tower be $h$ meters. Let the distance from Bob to the tower base be $x$ meters. Then the distance from Vicky to the tower base is $180 - x$ meters. 3. **Formulas:** Using the tangent of the angle of elevation: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{\text{distance}}$$ 4. **From Bob's position:** $$\tan(44^\circ) = \frac{h}{x} \implies h = x \tan(44^\circ)$$ 5. **From Vicky's position:** $$\tan(53^\circ) = \frac{h}{180 - x} \implies h = (180 - x) \tan(53^\circ)$$ 6. **Equate the two expressions for $h$:** $$x \tan(44^\circ) = (180 - x) \tan(53^\circ)$$ 7. **Solve for $x$:** $$x \tan(44^\circ) = 180 \tan(53^\circ) - x \tan(53^\circ)$$ $$x \tan(44^\circ) + x \tan(53^\circ) = 180 \tan(53^\circ)$$ $$x (\tan(44^\circ) + \tan(53^\circ)) = 180 \tan(53^\circ)$$ $$x = \frac{180 \tan(53^\circ)}{\tan(44^\circ) + \tan(53^\circ)}$$ 8. **Calculate values:** $$\tan(44^\circ) \approx 0.9657$$ $$\tan(53^\circ) \approx 1.3270$$ 9. **Substitute:** $$x = \frac{180 \times 1.3270}{0.9657 + 1.3270} = \frac{238.86}{2.2927} \approx 104.23$$ 10. **Find height $h$ using Bob's formula:** $$h = x \tan(44^\circ) = 104.23 \times 0.9657 \approx 100.7$$ **Final answer:** The height of the tower is approximately **100.7 meters**.