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 185√3 feet.
the distance from the man’s feet to the top of the monument is 370√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

We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) and \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\) and also for a \(30 - 60-90\) triangle, the ratios of sides are \(1:\sqrt{3}:2\) (opposite to \(30^{\circ}\), \(60^{\circ}\), \(90^{\circ}\) respectively).

First, let's find the adjacent side (distance from man's feet to base of monument, \(x\)) and hypotenuse (distance from man's feet to top of monument, \(y\)).

We know that \(\tan60^{\circ}=\sqrt{3}=\frac{h}{x}\), where \(h = 555\) feet. So \(x=\frac{h}{\tan60^{\circ}}=\frac{555}{\sqrt{3}}\). Rationalize the denominator: \(x=\frac{555\sqrt{3}}{3}=185\sqrt{3}\) feet.

Now, \(\sin60^{\circ}=\frac{\sqrt{3}}{2}=\frac{h}{y}\), so \(y = \frac{h}{\sin60^{\circ}}=\frac{555}{\frac{\sqrt{3}}{2}}=\frac{555\times2}{\sqrt{3}}=\frac{1110}{\sqrt{3}} = 370\sqrt{3}\) feet (after rationalizing \(\frac{1110\sqrt{3}}{3}=370\sqrt{3}\)). Also, we know that in a right - triangle, the hypotenuse is the longest side. The height of the monument is one of the legs. Since \(y = 370\sqrt{3}\approx370\times1.732 = 640.84\) feet and \(h = 555\) feet, and \(x = 185\sqrt{3}\approx185\times1.732=320.42\) feet. The hypotenuse \(y\) is longer than the height \(h\), so the statement "The segment representing the monument’s height is the longest segment in the triangle" is false.

Let's check each option:

• Option 1: The distance from the man’s feet to the base of the monument is \(185\sqrt{3}\) feet. From our calculation, \(x = 185\sqrt{3}\) feet. So this is correct.
• Option 2: The distance from the man’s feet to the top of the monument is \(370\sqrt{3}\) feet. From our calculation of \(y\), this is correct.
• Option 3: The distance from the man’s feet to the top of the monument is \(1110\) feet. \(370\sqrt{3}\approx370\times1.732 = 640.84

eq1110\), so this is incorrect.

• Option 4: The distance from the man’s feet to the base of the monument is \(277.5\) feet. \(185\sqrt{3}\approx320.42

eq277.5\), so this is incorrect.

• Option 5: The segment representing the monument’s height is the longest segment in the triangle. In a right - triangle, the hypotenuse is the longest side. The hypotenuse \(y = 370\sqrt{3}\approx640.84\) and the height of the monument \(h = 555\) feet. So the hypotenuse is longer than the height. So this statement is incorrect.

Answer:

The accurate measurements are:

• 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.