Question

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 man’s feet to the base of the monument is ( 185sqrt{3} ) feet.
□ the distance from the man’s feet to the top of the monument is ( 370sqrt{3} ) feet.
□ the distance from the man’s feet to the top of the monument is 1,110 feet.
□ the distance from the man’s feet to the base of the monument is 277.5 feet
□ the segment representing the monument’s height is the longest segment in the triangle

Explanation:

Step1: Identify the triangle type

We have a right - triangle where the height of the monument (opposite side to the \(60^{\circ}\) angle) \(h = 555\) feet, the angle of elevation \(\theta=60^{\circ}\), the distance from the man's feet to the base of the monument is the adjacent side (\(x\)) and the distance from the man's feet to the top of the monument is the hypotenuse (\(y\)).

Step2: Use trigonometric ratios

• For the adjacent side (\(x\)): We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), so \(\tan60^{\circ}=\frac{555}{x}\). Since \(\tan60^{\circ}=\sqrt{3}\), we have \(\sqrt{3}=\frac{555}{x}\), then \(x = \frac{555}{\sqrt{3}}=\frac{555\sqrt{3}}{3}=185\sqrt{3}\) feet.
• For the hypotenuse (\(y\)): We know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), so \(\sin60^{\circ}=\frac{555}{y}\). Since \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), we have \(\frac{\sqrt{3}}{2}=\frac{555}{y}\), then \(y=\frac{555\times2}{\sqrt{3}}=\frac{1110}{\sqrt{3}} = 370\sqrt{3}\) feet. Also, in a right - triangle, the hypotenuse is the longest side. The height of the monument is one of the legs (\(555\) feet), and the hypotenuse \(y = 370\sqrt{3}\approx370\times1.732 = 640.84\) feet which is greater than \(555\) feet. Let's check the third option: \(1110\) feet. Since \(y = 370\sqrt{3}\approx640.84

eq1110\), so the third option is wrong. The fourth option: \(x = 185\sqrt{3}\approx185\times1.732 = 320.42
eq277.5\), so the fourth option is wrong. The fifth option: The hypotenuse is the longest side, not the height (a leg), so the fifth option is wrong.

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.