Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Get quick and reliable solutions to your questions from a community of seasoned experts on our user-friendly platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
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]
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.