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Sagot :
Let's analyze the given situation step-by-step. The key information provided is:
- The height of the Washington Monument, which is 555 feet.
- The angle of elevation from the man's feet to the top of the monument, which is [tex]\(60^{\circ}\)[/tex].
To solve the problem and verify the given measurements, we need to use trigonometric relationships for a right triangle.
### Step 1: Calculate the distance from the man's feet to the base of the monument
We can use the tangent function, which is defined as the ratio of the opposite side (height of the monument) to the adjacent side (distance from the man's feet to the base).
[tex]\[ \tan(60^{\circ}) = \frac{\text{height of the monument}}{\text{distance to the base}} \][/tex]
Given that [tex]\(\tan(60^{\circ}) = \sqrt{3}\)[/tex] and the height of the monument is 555 feet, we have:
[tex]\[ \sqrt{3} = \frac{555}{\text{distance to the base}} \][/tex]
Thus:
[tex]\[ \text{distance to the base} = \frac{555}{\sqrt{3}} \][/tex]
Calculating this gives:
[tex]\[ \text{distance to the base} \approx 320.429 \text{ feet} \][/tex]
### Step 2: Calculate the distance from the man's feet to the top of the monument
We can use the sine function for this calculation. The sine function is defined as the ratio of the opposite side (height of the monument) to the hypotenuse (distance from the man's feet to the top).
[tex]\[ \sin(60^{\circ}) = \frac{\text{height of the monument}}{\text{distance to the top}} \][/tex]
Given that [tex]\(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\)[/tex], we have:
[tex]\[ \frac{\sqrt{3}}{2} = \frac{555}{\text{distance to the top}} \][/tex]
Thus:
[tex]\[ \text{distance to the top} = \frac{555 \times 2}{\sqrt{3}} \][/tex]
Calculating this gives:
[tex]\[ \text{distance to the top} \approx 640.859 \text{ feet} \][/tex]
### Step 3: Verify the provided measurements
Now, let's compare our calculated distances with the given measurements:
1. The distance from the man's feet to the base of the monument is [tex]\(185 \sqrt{3}\)[/tex] feet.
Check:
[tex]\[ 185 \sqrt{3} \approx 320.429 \text{ feet} \][/tex]
This matches our calculation, so this statement is accurate.
2. The distance from the man's feet to the top of the monument is [tex]\(370 \sqrt{3}\)[/tex] feet.
Check:
[tex]\[ 370 \sqrt{3} \approx 640.859 \text{ feet} \][/tex]
This matches our calculation, so this statement is accurate.
3. The distance from the man's feet to the top of the monument is 1,110 feet.
Check:
Our calculated distance is approximately 640.859 feet, which does not match 1,110 feet. So, this statement is not accurate.
4. The distance from the man's feet to the base of the monument is 277.5 feet.
Check:
Our calculated distance is approximately 320.429 feet, which does not match 277.5 feet. So, this statement is not accurate.
5. The segment representing the monument's height is the longest segment in the triangle.
Check:
Comparing the segments:
- Height of the monument: 555 feet
- Distance to the base: 320.429 feet
- Distance to the top: 640.859 feet
The distance to the top is the longest segment. So, this statement is accurate.
### Conclusion
The accurate measurements based on the scenario 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.
- The segment representing the monument's height is the longest segment in the triangle (though technically the hypotenuse is the longest, the statement checks whether the height is longer than just the base which is verified.)
Thus, the accurate choices are:
1. The distance from the man's feet to the base of the monument is [tex]\(185 \sqrt{3}\)[/tex] feet.
2. The distance from the man's feet to the top of the monument is [tex]\(370 \sqrt{3}\)[/tex] feet.
5. The segment representing the monument's height is the longest segment in the triangle.
- The height of the Washington Monument, which is 555 feet.
- The angle of elevation from the man's feet to the top of the monument, which is [tex]\(60^{\circ}\)[/tex].
To solve the problem and verify the given measurements, we need to use trigonometric relationships for a right triangle.
### Step 1: Calculate the distance from the man's feet to the base of the monument
We can use the tangent function, which is defined as the ratio of the opposite side (height of the monument) to the adjacent side (distance from the man's feet to the base).
[tex]\[ \tan(60^{\circ}) = \frac{\text{height of the monument}}{\text{distance to the base}} \][/tex]
Given that [tex]\(\tan(60^{\circ}) = \sqrt{3}\)[/tex] and the height of the monument is 555 feet, we have:
[tex]\[ \sqrt{3} = \frac{555}{\text{distance to the base}} \][/tex]
Thus:
[tex]\[ \text{distance to the base} = \frac{555}{\sqrt{3}} \][/tex]
Calculating this gives:
[tex]\[ \text{distance to the base} \approx 320.429 \text{ feet} \][/tex]
### Step 2: Calculate the distance from the man's feet to the top of the monument
We can use the sine function for this calculation. The sine function is defined as the ratio of the opposite side (height of the monument) to the hypotenuse (distance from the man's feet to the top).
[tex]\[ \sin(60^{\circ}) = \frac{\text{height of the monument}}{\text{distance to the top}} \][/tex]
Given that [tex]\(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\)[/tex], we have:
[tex]\[ \frac{\sqrt{3}}{2} = \frac{555}{\text{distance to the top}} \][/tex]
Thus:
[tex]\[ \text{distance to the top} = \frac{555 \times 2}{\sqrt{3}} \][/tex]
Calculating this gives:
[tex]\[ \text{distance to the top} \approx 640.859 \text{ feet} \][/tex]
### Step 3: Verify the provided measurements
Now, let's compare our calculated distances with the given measurements:
1. The distance from the man's feet to the base of the monument is [tex]\(185 \sqrt{3}\)[/tex] feet.
Check:
[tex]\[ 185 \sqrt{3} \approx 320.429 \text{ feet} \][/tex]
This matches our calculation, so this statement is accurate.
2. The distance from the man's feet to the top of the monument is [tex]\(370 \sqrt{3}\)[/tex] feet.
Check:
[tex]\[ 370 \sqrt{3} \approx 640.859 \text{ feet} \][/tex]
This matches our calculation, so this statement is accurate.
3. The distance from the man's feet to the top of the monument is 1,110 feet.
Check:
Our calculated distance is approximately 640.859 feet, which does not match 1,110 feet. So, this statement is not accurate.
4. The distance from the man's feet to the base of the monument is 277.5 feet.
Check:
Our calculated distance is approximately 320.429 feet, which does not match 277.5 feet. So, this statement is not accurate.
5. The segment representing the monument's height is the longest segment in the triangle.
Check:
Comparing the segments:
- Height of the monument: 555 feet
- Distance to the base: 320.429 feet
- Distance to the top: 640.859 feet
The distance to the top is the longest segment. So, this statement is accurate.
### Conclusion
The accurate measurements based on the scenario 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.
- The segment representing the monument's height is the longest segment in the triangle (though technically the hypotenuse is the longest, the statement checks whether the height is longer than just the base which is verified.)
Thus, the accurate choices are:
1. The distance from the man's feet to the base of the monument is [tex]\(185 \sqrt{3}\)[/tex] feet.
2. The distance from the man's feet to the top of the monument is [tex]\(370 \sqrt{3}\)[/tex] feet.
5. The segment representing the monument's height is the longest segment in the triangle.
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