Let's analyze the given information and verify which measurements are accurate.
Given:
- The height of the monument is 555 feet.
- Distance from the man’s feet to the base of the monument could be [tex]$185\sqrt{3}$[/tex] feet.
To Validate:
1. Distance from the man's feet to the base of the monument being [tex]$185\sqrt{3}$[/tex] feet:
Given distance to base is indeed [tex]$185\sqrt{3}$[/tex] feet. This is correct and thus:
[tex]$\mathbf{\surd} \text{The distance from the man's feet to the base of the monument is } 185\sqrt{3} \text{ feet.}$[/tex]
2. Distance from the man's feet to the top of the monument being [tex]$370\sqrt{3}$[/tex] feet:
Using the Pythagorean theorem to calculate distance from feet to top of the monument, where the distance to the base [tex]$185\sqrt{3}$[/tex]:
[tex]\[ \text{Distance to the top} = \sqrt{(555)^2 + (185\sqrt{3})^2} \][/tex]
The value [tex]$ 370\sqrt{3}$[/tex] for the distance from the man's feet to the top does match our calculation:
[tex]$\mathbf{\surd} \text{The distance from the man's feet to the top of the monument is } 370\sqrt{3} \text{ feet.}$[/tex]
3. Distance from the man's feet to the top of the monument being 1,110 feet:
Using the Pythagorean theorem calculation would not yield 1,110 feet because:
[tex]\[ \sqrt{(555)^2 + (185 \sqrt{3})^2} < 1110 \][/tex]
Thus the distance to the top is not correct being 1110 feet:
[tex]$\square \text{The distance from the man's feet to the top of the monument is 1,110 feet.}$[/tex]
4. Distance from the man's feet to the base of the monument is 277.5 feet:
Given distance to the base [tex]$185\sqrt{3}$[/tex] does not match 277.5 feet.
[tex]\[ 185\sqrt{3} \neq 277.5 \][/tex]
Thus the distance to the base is not accurate as 277.5 feet:
[tex]$\square \text{The distance from the man's feet to the base of the monument is 277.5 feet.}$[/tex]
5. The segment representing the monument’s height is the longest segment in the triangle:
Comparing distances between height of the monument (555 feet), base distance ([tex]$185\sqrt{3}$[/tex] feet), and top distance:
[tex]\[ 555 > 370\sqrt{3} \rightarrow \text{False} \][/tex]
Thus, the segment representing height is not the longest:
[tex]$\square \text{The segment representing the monument's height is the longest segment in the triangle.}$[/tex]
Thus, the accurate measurements are:
- The distance from the man's feet to the base of the monument is [tex]$185\sqrt{3}$[/tex] feet.
- The distance from the man's feet to the top of the monument is [tex]$370\sqrt{3}$[/tex] feet.
Concluding, check:
[tex]$\mathbf{\surd} \text{The distance from the man's feet to the base of the monument is } 185\sqrt{3} \text{ feet.}$[/tex]
[tex]$\mathbf{\surd} \text{The distance from the man's feet to the top of the monument is } 370\sqrt{3} \text{ feet.}$[/tex]
