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Sagot :
Certainly! Let's work through this problem step-by-step to find the distance [tex]\( x \)[/tex] the ladder is moved towards the wall, rounded to the nearest meter.
1. Understanding the Problem:
- Initially, the top of the ladder is 10 meters from the ground when it leans against a wall, making an angle of [tex]\( 35.5^\circ \)[/tex] with respect to the ground.
- After the ladder is moved closer to the wall by a distance [tex]\( x \)[/tex], it creates an angle of [tex]\( 54.5^\circ \)[/tex] with the ground, and its top is then 14 meters above the ground.
- We need to find the value of [tex]\( x \)[/tex].
2. Step-by-Step Solution:
- First, we need to find the horizontal distances from the wall (the base of the ladder) in both initial and final positions.
- Let’s denote:
- [tex]\( h_1 = 10 \)[/tex] meters (initial height)
- [tex]\( \theta_1 = 35.5^\circ \)[/tex] (initial angle)
- [tex]\( h_2 = 14 \)[/tex] meters (final height)
- [tex]\( \theta_2 = 54.5^\circ \)[/tex] (final angle)
3. Calculate the Initial Horizontal Distance:
We use the trigonometric relation:
[tex]\[ \tan(\theta) = \frac{\text{height}}{\text{base}} \][/tex]
For the initial position:
[tex]\[ \tan(35.5^\circ) = \frac{10}{d_1} \][/tex]
Where [tex]\( d_1 \)[/tex] is the initial horizontal distance.
[tex]\[ d_1 = \frac{10}{\tan(35.5^\circ)} \][/tex]
4. Calculate the Final Horizontal Distance:
For the final position:
[tex]\[ \tan(54.5^\circ) = \frac{14}{d_2} \][/tex]
Where [tex]\( d_2 \)[/tex] is the final horizontal distance.
[tex]\[ d_2 = \frac{14}{\tan(54.5^\circ)} \][/tex]
5. Calculate the Distance Moved:
The distance the ladder is moved towards the wall is [tex]\( x \)[/tex], which is the difference between the initial and final horizontal distances:
[tex]\[ x = d_1 - d_2 \][/tex]
6. Rounding to the Nearest Meter:
After calculating the above steps (finding [tex]\( d_1 \)[/tex] and [tex]\( d_2 \)[/tex]), the difference [tex]\( x \)[/tex] is found, and it should be rounded to the nearest meter.
Based on the calculations:
[tex]\[ x \approx 4 \][/tex]
Therefore, the distance the ladder is moved towards the wall, rounded to the nearest meter, is:
[tex]\[ \text{Answer: } x = 4 \text{ meters} \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{4 \text{ meters}} \][/tex]
1. Understanding the Problem:
- Initially, the top of the ladder is 10 meters from the ground when it leans against a wall, making an angle of [tex]\( 35.5^\circ \)[/tex] with respect to the ground.
- After the ladder is moved closer to the wall by a distance [tex]\( x \)[/tex], it creates an angle of [tex]\( 54.5^\circ \)[/tex] with the ground, and its top is then 14 meters above the ground.
- We need to find the value of [tex]\( x \)[/tex].
2. Step-by-Step Solution:
- First, we need to find the horizontal distances from the wall (the base of the ladder) in both initial and final positions.
- Let’s denote:
- [tex]\( h_1 = 10 \)[/tex] meters (initial height)
- [tex]\( \theta_1 = 35.5^\circ \)[/tex] (initial angle)
- [tex]\( h_2 = 14 \)[/tex] meters (final height)
- [tex]\( \theta_2 = 54.5^\circ \)[/tex] (final angle)
3. Calculate the Initial Horizontal Distance:
We use the trigonometric relation:
[tex]\[ \tan(\theta) = \frac{\text{height}}{\text{base}} \][/tex]
For the initial position:
[tex]\[ \tan(35.5^\circ) = \frac{10}{d_1} \][/tex]
Where [tex]\( d_1 \)[/tex] is the initial horizontal distance.
[tex]\[ d_1 = \frac{10}{\tan(35.5^\circ)} \][/tex]
4. Calculate the Final Horizontal Distance:
For the final position:
[tex]\[ \tan(54.5^\circ) = \frac{14}{d_2} \][/tex]
Where [tex]\( d_2 \)[/tex] is the final horizontal distance.
[tex]\[ d_2 = \frac{14}{\tan(54.5^\circ)} \][/tex]
5. Calculate the Distance Moved:
The distance the ladder is moved towards the wall is [tex]\( x \)[/tex], which is the difference between the initial and final horizontal distances:
[tex]\[ x = d_1 - d_2 \][/tex]
6. Rounding to the Nearest Meter:
After calculating the above steps (finding [tex]\( d_1 \)[/tex] and [tex]\( d_2 \)[/tex]), the difference [tex]\( x \)[/tex] is found, and it should be rounded to the nearest meter.
Based on the calculations:
[tex]\[ x \approx 4 \][/tex]
Therefore, the distance the ladder is moved towards the wall, rounded to the nearest meter, is:
[tex]\[ \text{Answer: } x = 4 \text{ meters} \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{4 \text{ meters}} \][/tex]
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