Tower Height 1F3E9F

1. **State the problem:** We have a vertical tower with a flagstaff of length $10$ m on top. From a point on level ground, the angle of elevation to the top of the tower is $45^\circ$ and to the top of the flagstaff is $60^\circ$. We need to find the height of the tower and the horizontal distance from the point to the base of the tower. 2. **Define variables:** Let the height of the tower be $h$ meters. Let the horizontal distance from the point to the base of the tower be $d$ meters. 3. **Use trigonometric relations:** - The angle of elevation to the top of the tower is $45^\circ$, so: $$\tan 45^\circ = \frac{h}{d}$$ Since $\tan 45^\circ = 1$, we get: $$1 = \frac{h}{d} \implies h = d$$ - The angle of elevation to the top of the flagstaff (which is $h + 10$ meters high) is $60^\circ$, so: $$\tan 60^\circ = \frac{h + 10}{d}$$ Since $\tan 60^\circ = \sqrt{3}$, we get: $$\sqrt{3} = \frac{h + 10}{d}$$ 4. **Substitute $h = d$ into the second equation:** $$\sqrt{3} = \frac{d + 10}{d} = 1 + \frac{10}{d}$$ 5. **Solve for $d$:** $$\sqrt{3} - 1 = \frac{10}{d}$$ $$d = \frac{10}{\sqrt{3} - 1}$$ 6. **Rationalize the denominator:** $$d = \frac{10}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = \frac{10(\sqrt{3} + 1)}{(\sqrt{3})^2 - 1^2} = \frac{10(\sqrt{3} + 1)}{3 - 1} = \frac{10(\sqrt{3} + 1)}{2}$$ 7. **Calculate exact height $h$:** Since $h = d$, we have: $$h = \frac{10(\sqrt{3} + 1)}{2}$$ 8. **Calculate decimal approximations:** $$\sqrt{3} \approx 1.732$$ $$h \approx \frac{10(1.732 + 1)}{2} = \frac{10(2.732)}{2} = 5 \times 2.732 = 13.66$$ So, - Height of the tower $h \approx 13.66$ m - Distance from the point $d \approx 13.66$ m **Final answers:** - Exact height of the tower: $$h = \frac{10(\sqrt{3} + 1)}{2}$$ meters - Distance from the point: $$d = \frac{10(\sqrt{3} + 1)}{2}$$ meters - Height to 2 decimal places: $13.66$ meters - Distance to 2 decimal places: $13.66$ meters