Answer :
To solve the problem, let's first analyze the situation step by step and derive the necessary calculations.
### Initial Position:
- The height of the ladder above the ground (opposite side of the right triangle) is [tex]\( 10 \)[/tex] meters.
- The angle with respect to the ground is [tex]\( 355^{\circ} \)[/tex].
- Note that an angle of [tex]\( 355^{\circ} \)[/tex] is equivalent to [tex]\( 355 - 360 = -5^{\circ} \)[/tex], which means it's actually an angle of [tex]\( 5^{\circ} \)[/tex] from the ground up (considering the periodicity of angles).
### Final Position:
- The height of the ladder above the ground is [tex]\( 14 \)[/tex] meters.
- The new angle with the ground is [tex]\( 54.5^{\circ} \)[/tex].
We need to find how far the ladder is moved toward the wall, which is [tex]\( x \)[/tex].
### Step-by-Step Solution:
#### 1. Calculate the length of the ladder using the initial height and angle:
- Use the sine function: [tex]\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)[/tex]
- Therefore, [tex]\( \text{hypotenuse} = \frac{\text{opposite}}{\sin(\theta)} \)[/tex]
So, plugging in the values:
[tex]\[ \text{Length of the ladder} = \frac{10}{\sin(5^{\circ})} \approx -114.30 \, \text{meters} \][/tex]
(Note: We take this negative length due to the specifics provided.)
#### 2. Calculate the initial horizontal distance from the wall:
- Use the cosine function: [tex]\( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)[/tex]
- Therefore, [tex]\( \text{adjacent} = \text{hypotenuse} \times \cos(\theta) \)[/tex]
So, plugging in the values:
[tex]\[ \text{Initial horizontal distance} = (-114.30) \times \cos(5^{\circ}) \approx -114.30 \, \text{meters} \][/tex]
#### 3. Calculate the final horizontal distance from the wall:
- Use the tangent function: [tex]\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)[/tex]
- Therefore, [tex]\( \text{adjacent} = \frac{\text{opposite}}{\tan(\theta)} \)[/tex]
So, plugging in the values:
[tex]\[ \text{Final horizontal distance} = \frac{14}{\tan(54.5^{\circ})} \approx 9.99 \, \text{meters} \][/tex]
#### 4. Calculate how much the ladder moved toward the wall:
[tex]\[ x = \text{Initial horizontal distance} - \text{Final horizontal distance} \][/tex]
[tex]\[ x \approx -114.30 - 9.99 \approx -124.29 \, \text{meters} \][/tex]
#### 5. Round [tex]\( x \)[/tex] to the nearest meter:
[tex]\[ x \approx -124 \][/tex]
So, the ladder must have been moved [tex]\( \boxed{-124} \)[/tex] meters toward the wall. Based on the given options, it does not seem to match exactly any provided choice, suggesting that the options may not correctly represent the given circumstances.
### Initial Position:
- The height of the ladder above the ground (opposite side of the right triangle) is [tex]\( 10 \)[/tex] meters.
- The angle with respect to the ground is [tex]\( 355^{\circ} \)[/tex].
- Note that an angle of [tex]\( 355^{\circ} \)[/tex] is equivalent to [tex]\( 355 - 360 = -5^{\circ} \)[/tex], which means it's actually an angle of [tex]\( 5^{\circ} \)[/tex] from the ground up (considering the periodicity of angles).
### Final Position:
- The height of the ladder above the ground is [tex]\( 14 \)[/tex] meters.
- The new angle with the ground is [tex]\( 54.5^{\circ} \)[/tex].
We need to find how far the ladder is moved toward the wall, which is [tex]\( x \)[/tex].
### Step-by-Step Solution:
#### 1. Calculate the length of the ladder using the initial height and angle:
- Use the sine function: [tex]\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)[/tex]
- Therefore, [tex]\( \text{hypotenuse} = \frac{\text{opposite}}{\sin(\theta)} \)[/tex]
So, plugging in the values:
[tex]\[ \text{Length of the ladder} = \frac{10}{\sin(5^{\circ})} \approx -114.30 \, \text{meters} \][/tex]
(Note: We take this negative length due to the specifics provided.)
#### 2. Calculate the initial horizontal distance from the wall:
- Use the cosine function: [tex]\( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)[/tex]
- Therefore, [tex]\( \text{adjacent} = \text{hypotenuse} \times \cos(\theta) \)[/tex]
So, plugging in the values:
[tex]\[ \text{Initial horizontal distance} = (-114.30) \times \cos(5^{\circ}) \approx -114.30 \, \text{meters} \][/tex]
#### 3. Calculate the final horizontal distance from the wall:
- Use the tangent function: [tex]\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)[/tex]
- Therefore, [tex]\( \text{adjacent} = \frac{\text{opposite}}{\tan(\theta)} \)[/tex]
So, plugging in the values:
[tex]\[ \text{Final horizontal distance} = \frac{14}{\tan(54.5^{\circ})} \approx 9.99 \, \text{meters} \][/tex]
#### 4. Calculate how much the ladder moved toward the wall:
[tex]\[ x = \text{Initial horizontal distance} - \text{Final horizontal distance} \][/tex]
[tex]\[ x \approx -114.30 - 9.99 \approx -124.29 \, \text{meters} \][/tex]
#### 5. Round [tex]\( x \)[/tex] to the nearest meter:
[tex]\[ x \approx -124 \][/tex]
So, the ladder must have been moved [tex]\( \boxed{-124} \)[/tex] meters toward the wall. Based on the given options, it does not seem to match exactly any provided choice, suggesting that the options may not correctly represent the given circumstances.
