Certainly! Let's walk through the details of solving this problem step-by-step.
### Problem Recap
We have a ladder positioned against a pipe, forming a right-angled triangle with the ground. Given:
- The angle where the ladder meets the ground is [tex]\( 56.4^\circ \)[/tex].
- The ladder, the ground, and the pipe form a right triangle.
To find: The angle at which the pipe meets the ground.
### Step-by-Step Solution
1. Identify the Angle Types in the Triangle:
- The angle between the ladder and the ground: [tex]\( 56.4^\circ \)[/tex].
- The angle between the ground and the pipe, denoted as [tex]\( \theta \)[/tex].
- The right angle formed between the pipe and the ground: [tex]\( 90^\circ \)[/tex].
Since these angles sum up to [tex]\( 180^\circ \)[/tex] (the property of a triangle):
2. Relate the Known and Unknown Angles:
- We know the three angles must add up to [tex]\( 180^\circ \)[/tex]:
[tex]\[ \angle_{\text{ladder-ground}} + \angle_{\text{ladder-pipe}} + \angle_{\text{ground-pipe}} = 180^\circ \][/tex]
[tex]\[ 56.4^\circ + 90^\circ + \theta = 180^\circ \][/tex]
3. Solve for the Unknown Angle [tex]\(\theta\)[/tex]:
- Rearrange the equation to solve for [tex]\(\theta\)[/tex]:
[tex]\[ \theta = 180^\circ - 90^\circ - 56.4^\circ \][/tex]
[tex]\[ \theta = 90^\circ - 56.4^\circ \][/tex]
[tex]\[ \theta = 33.6^\circ \][/tex]
### Answer
The angle at which the pipe meets the ground is [tex]\( 33.6^\circ \)[/tex], which is not among the provided choices, but it is indeed the correct solution based on the problem statement.
