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Sagot :
Absolutely! Let's go through the problem step-by-step to find the height of the statue:
1. Understand the Problem:
- A man is standing such that the line from his head to the top of a statue makes a 26° angle with the horizontal.
- This line (diagonal distance) from his head to the top of the statue is 57 feet long.
- The height of the man is 6 feet.
- We need to find the total height of the statue.
2. Break Down the Problem:
- Imagine a right triangle where:
- The hypotenuse is the diagonal distance of 57 feet.
- The angle of elevation at the man’s eye level (head) to the top of the statue is 26°.
- We need to find the vertical height from the man's head to the top of the statue (opposite side of the angle).
3. Calculate the Vertical Height Component:
- To find the vertical height component ([tex]\(h\)[/tex]) of the statue from the top of the man's head, we use the sine function of the angle of elevation. Remember that in a right triangle, [tex]\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)[/tex].
Thus,
[tex]\[ \sin(26^\circ) = \frac{\text{vertical height from head to statue top}}{57 \text{ feet}} \][/tex]
[tex]\[ \sin(26^\circ) = \frac{h}{57} \][/tex]
[tex]\[ h = 57 \times \sin(26^\circ) \][/tex]
4. Converting the Angle to Radians:
- Since we are dealing with trigonometric functions, it's easier to work in radians:
[tex]\[ 26^\circ = 0.4537856055185257 \text{ radians} \][/tex]
5. Finding the Vertical Height:
- Substitute [tex]\( \sin(26^\circ) \)[/tex] with its value:
[tex]\[ h = 57 \times \sin(0.4537856055185257) \][/tex]
[tex]\[ h = 57 \times 0.4383711467890774 \][/tex]
[tex]\[ h \approx 24.98715536697741 \text{ feet} \][/tex]
6. Total Height of the Statue:
- Now, add the height of the man (6 feet) to the vertical height component calculated above to get the total height of the statue:
[tex]\[ \text{Total height of the statue} = 24.98715536697741 \text{ feet} + 6 \text{ feet} \][/tex]
[tex]\[ \text{Total height of the statue} \approx 30.98715536697741 \text{ feet} \][/tex]
So, the total height of the statue is approximately 30.99 feet.
1. Understand the Problem:
- A man is standing such that the line from his head to the top of a statue makes a 26° angle with the horizontal.
- This line (diagonal distance) from his head to the top of the statue is 57 feet long.
- The height of the man is 6 feet.
- We need to find the total height of the statue.
2. Break Down the Problem:
- Imagine a right triangle where:
- The hypotenuse is the diagonal distance of 57 feet.
- The angle of elevation at the man’s eye level (head) to the top of the statue is 26°.
- We need to find the vertical height from the man's head to the top of the statue (opposite side of the angle).
3. Calculate the Vertical Height Component:
- To find the vertical height component ([tex]\(h\)[/tex]) of the statue from the top of the man's head, we use the sine function of the angle of elevation. Remember that in a right triangle, [tex]\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)[/tex].
Thus,
[tex]\[ \sin(26^\circ) = \frac{\text{vertical height from head to statue top}}{57 \text{ feet}} \][/tex]
[tex]\[ \sin(26^\circ) = \frac{h}{57} \][/tex]
[tex]\[ h = 57 \times \sin(26^\circ) \][/tex]
4. Converting the Angle to Radians:
- Since we are dealing with trigonometric functions, it's easier to work in radians:
[tex]\[ 26^\circ = 0.4537856055185257 \text{ radians} \][/tex]
5. Finding the Vertical Height:
- Substitute [tex]\( \sin(26^\circ) \)[/tex] with its value:
[tex]\[ h = 57 \times \sin(0.4537856055185257) \][/tex]
[tex]\[ h = 57 \times 0.4383711467890774 \][/tex]
[tex]\[ h \approx 24.98715536697741 \text{ feet} \][/tex]
6. Total Height of the Statue:
- Now, add the height of the man (6 feet) to the vertical height component calculated above to get the total height of the statue:
[tex]\[ \text{Total height of the statue} = 24.98715536697741 \text{ feet} + 6 \text{ feet} \][/tex]
[tex]\[ \text{Total height of the statue} \approx 30.98715536697741 \text{ feet} \][/tex]
So, the total height of the statue is approximately 30.99 feet.
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