To solve this problem, we need to follow a series of trigonometry steps to determine the horizontal displacement [tex]\(x\)[/tex] of the ladder.
Firstly, recall the two given heights of the ladder and their respective angles with the ground:
- Initially, the top of the ladder is 10 meters from the ground with an angle of [tex]\(35.5^{\circ}\)[/tex].
- Finally, the top of the ladder is 14 meters from the ground with an angle of [tex]\(54.5^{\circ}\)[/tex].
Step 1: Calculate the horizontal distance from the wall where the ladder touches the ground in both scenarios.
Initial position:
- Height [tex]\( h_1 = 10 \)[/tex] meters.
- Angle [tex]\( \theta_1 = 35.5^{\circ} \)[/tex].
The horizontal base ([tex]\( b_1 \)[/tex]) can be found using the tangent function:
[tex]\[
\tan(\theta_1) = \frac{h_1}{b_1}
\][/tex]
Solving for [tex]\( b_1 \)[/tex]:
[tex]\[
b_1 = \frac{h_1}{\tan(\theta_1)} = \frac{10}{\tan(35.5^{\circ})} \approx 14.01948294476336 \text{ meters}
\][/tex]
Final position:
- Height [tex]\( h_2 = 14 \)[/tex] meters.
- Angle [tex]\( \theta_2 = 54.5^{\circ} \)[/tex].
The horizontal base ([tex]\( b_2 \)[/tex]) can likewise be found:
[tex]\[
\tan(\theta_2) = \frac{h_2}{b_2}
\][/tex]
Solving for [tex]\( b_2 \)[/tex]:
[tex]\[
b_2 = \frac{h_2}{\tan(\theta_2)} = \frac{14}{\tan(54.5^{\circ})} \approx 9.986102950558077 \text{ meters}
\][/tex]
Step 2: Calculate the distance [tex]\( x \)[/tex] the ladder moved towards the wall.
The horizontal distance moved horizontally [tex]\( x \)[/tex] is the difference between [tex]\( b_1 \)[/tex] and [tex]\( b_2 \)[/tex]:
[tex]\[
x = b_1 - b_2 \approx 14.01948294476336 - 9.986102950558077 \approx 4.0333799942052835 \text{ meters}
\][/tex]
Step 3: Round [tex]\( x \)[/tex] to the nearest meter.
[tex]\[
x \approx 4 \text{ meters}
\][/tex]
Therefore, the ladder moved approximately 4 meters towards the wall. Hence, the correct answer is:
B. 4 meters
