Question

150 ft
45° 60°
not drawn to scale
what is the distance from the first observation point to the top of the tower (x)? round the answer to the nearest tenth.
122.5 ft
183.7 ft

Explanation:

Step1: Find the height of the tower (h) using the 60° triangle

In the right triangle with angle 60° and hypotenuse 150 ft, we use the sine function: $\sin(60^\circ) = \frac{h}{150}$. So, $h = 150 \times \sin(60^\circ) = 150 \times \frac{\sqrt{3}}{2} = 75\sqrt{3}$ ft.

Step2: Find x using the 45° triangle

In the right triangle with angle 45° and height h, since $\sin(45^\circ) = \frac{h}{x}$, we can solve for x: $x = \frac{h}{\sin(45^\circ)}$. Substitute $h = 75\sqrt{3}$: $x = \frac{75\sqrt{3}}{\frac{\sqrt{2}}{2}} = 75\sqrt{3} \times \frac{2}{\sqrt{2}} = 75\sqrt{6}$. Calculate the numerical value: $\sqrt{6} \approx 2.449$, so $x \approx 75 \times 2.449 \approx 183.7$ ft.

Answer:

183.7 ft (Option: 183.7 ft)