Answer :
To solve this problem, we need to calculate the horizontal distances of the ladder from the wall for both given angles and subtract these distances to find the horizontal distance [tex]\( x \)[/tex] that the ladder was moved. These steps will be clearly outlined below:
1. Determine the horizontal distance from the wall at the first angle:
- The height from the ground to the top of the ladder when it is leaning at [tex]\(35.5^\circ\)[/tex] is 10 meters.
- We use trigonometry to find the horizontal distance (base) of the ladder from the wall. Specifically, we use the tangent function:
[tex]\[ \tan(\theta_1) = \frac{\text{opposite}}{\text{adjacent}} \Rightarrow \text{adjacent} = \frac{\text{opposite}}{\tan(\theta_1)} \][/tex]
Here, the opposite side is the height (10 meters) and [tex]\(\theta_1\)[/tex] is the angle (35.5 degrees):
[tex]\[ \text{Base}_1 = \frac{10}{\tan(35.5^\circ)} \][/tex]
From the calculations,
[tex]\[ \text{Base}_1 \approx 14.02 \text{ meters} \][/tex]
2. Determine the horizontal distance from the wall at the second angle:
- The height from the ground to the top of the ladder when it is leaning at [tex]\(54.5^\circ\)[/tex] is 14 meters.
- Again, we use the tangent function to find the horizontal distance of the ladder from the wall:
[tex]\[ \tan(\theta_2) = \frac{\text{opposite}}{\text{adjacent}} \Rightarrow \text{adjacent} = \frac{\text{opposite}}{\tan(\theta_2)} \][/tex]
In this case, the opposite side is 14 meters and [tex]\(\theta_2\)[/tex] is the angle (54.5 degrees):
[tex]\[ \text{Base}_2 = \frac{14}{\tan(54.5^\circ)} \][/tex]
From the calculations,
[tex]\[ \text{Base}_2 \approx 9.99 \text{ meters} \][/tex]
3. Calculate the distance [tex]\( x \)[/tex] the ladder was moved:
- To find [tex]\( x \)[/tex], we subtract the second horizontal distance from the first horizontal distance:
[tex]\[ x = \text{Base}_1 - \text{Base}_2 \][/tex]
From the calculations,
[tex]\[ x \approx 14.02 - 9.99 \approx 4.03 \text{ meters} \][/tex]
4. Round [tex]\( x \)[/tex] to the nearest meter:
[tex]\[ \text{Rounded } x = 4 \text{ meters} \][/tex]
So, the ladder was moved approximately 4 meters toward the wall. Therefore, the correct answer is:
D. 4 meters
1. Determine the horizontal distance from the wall at the first angle:
- The height from the ground to the top of the ladder when it is leaning at [tex]\(35.5^\circ\)[/tex] is 10 meters.
- We use trigonometry to find the horizontal distance (base) of the ladder from the wall. Specifically, we use the tangent function:
[tex]\[ \tan(\theta_1) = \frac{\text{opposite}}{\text{adjacent}} \Rightarrow \text{adjacent} = \frac{\text{opposite}}{\tan(\theta_1)} \][/tex]
Here, the opposite side is the height (10 meters) and [tex]\(\theta_1\)[/tex] is the angle (35.5 degrees):
[tex]\[ \text{Base}_1 = \frac{10}{\tan(35.5^\circ)} \][/tex]
From the calculations,
[tex]\[ \text{Base}_1 \approx 14.02 \text{ meters} \][/tex]
2. Determine the horizontal distance from the wall at the second angle:
- The height from the ground to the top of the ladder when it is leaning at [tex]\(54.5^\circ\)[/tex] is 14 meters.
- Again, we use the tangent function to find the horizontal distance of the ladder from the wall:
[tex]\[ \tan(\theta_2) = \frac{\text{opposite}}{\text{adjacent}} \Rightarrow \text{adjacent} = \frac{\text{opposite}}{\tan(\theta_2)} \][/tex]
In this case, the opposite side is 14 meters and [tex]\(\theta_2\)[/tex] is the angle (54.5 degrees):
[tex]\[ \text{Base}_2 = \frac{14}{\tan(54.5^\circ)} \][/tex]
From the calculations,
[tex]\[ \text{Base}_2 \approx 9.99 \text{ meters} \][/tex]
3. Calculate the distance [tex]\( x \)[/tex] the ladder was moved:
- To find [tex]\( x \)[/tex], we subtract the second horizontal distance from the first horizontal distance:
[tex]\[ x = \text{Base}_1 - \text{Base}_2 \][/tex]
From the calculations,
[tex]\[ x \approx 14.02 - 9.99 \approx 4.03 \text{ meters} \][/tex]
4. Round [tex]\( x \)[/tex] to the nearest meter:
[tex]\[ \text{Rounded } x = 4 \text{ meters} \][/tex]
So, the ladder was moved approximately 4 meters toward the wall. Therefore, the correct answer is:
D. 4 meters
