Question

solving real - world problems with 30° - 60° - 90° triangles
a man is standing near the washington monument. at a 60° angle of elevation from the ground, the man sees the top of the 555 - foot monument.
which measurements are accurate based on the scenario? check all that apply.
□ the distance from the mans feet to the base of the monument is 185√3 feet.
□ the distance from the man’s feet to the top of the monument is 370√3 feet.
□ the distance from the mans feet to the top of the monument is 1,110 feet.
□ the distance from the mans feet to the base of the monument is 277.5 feet.
□ the segment representing the monuments height is the longest segment in the triangle.

Explanation:

Step1: Define triangle sides

Let:

• Height (opposite 60°): $h = 555$ ft
• Adjacent (distance to base): $x$
• Hypotenuse (distance to top): $y$

Step2: Solve for adjacent side

Use $\tan(60^\circ)=\sqrt{3}=\frac{h}{x}$
$x = \frac{555}{\sqrt{3}} = \frac{555\sqrt{3}}{3} = 185\sqrt{3}$ ft

Step3: Solve for hypotenuse

Use $\sin(60^\circ)=\frac{\sqrt{3}}{2}=\frac{h}{y}$
$y = \frac{555 \times 2}{\sqrt{3}} = \frac{1110\sqrt{3}}{3} = 370\sqrt{3}$ ft

Step4: Verify longest segment

Hypotenuse $y=370\sqrt{3}\approx641$ ft, which is longer than 555 ft.

Answer:

• The distance from the man's feet to the base of the monument is $185\sqrt{3}$ feet.
• The distance from the man's feet to the top of the monument is $370\sqrt{3}$ feet.