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 mans feet to the base of the monument is 185√3 feet. the distance from the mans 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: Find the distance from the man's feet to the base of the monument

Let \(x\) be the distance from the man's feet to the base of the monument.
We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), where \(\theta = 60^{\circ}\) and the opposite side is the height of the monument \(h = 555\) feet.
\(\tan60^{\circ}=\frac{555}{x}\)
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.

Step2: Find the distance from the man's feet to the top of the monument

Let \(y\) be the distance from the man's feet to the top of the monument.
We know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\theta = 60^{\circ}\), opposite side \(h = 555\) feet.
\(\sin60^{\circ}=\frac{555}{y}\)
Since \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), we have \(y=\frac{555\times2}{\sqrt{3}}=\frac{1110\sqrt{3}}{3} = 370\sqrt{3}\) feet.

Step3: Analyze the length of the sides in the right - triangle

In a right - triangle, the hypotenuse is the longest side. The height of the monument (\(555\) feet) is not the hypotenuse. The hypotenuse (distance from man's feet to the top of the monument \(y = 370\sqrt{3}\approx370\times1.732 = 640.84\) feet) is longer than the height of the monument. Also, \(277.5
eq185\sqrt{3}\approx185\times1.732 = 319.42\) and \(1110
eq370\sqrt{3}\approx640.84\)

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.