Two vertical posts stand side by side. One post is 8 feet tall and the other is 17 feet tall. If a 24-foot wire is stretched between the tops of the posts, how far apart are the posts?
The correct answer and explanation is :
To solve this problem, we will treat it as a geometry problem involving the concept of right triangles. Let’s break it down step by step.
Step 1: Visualizing the Problem
We have two vertical posts, one 8 feet tall and the other 17 feet tall. A 24-foot wire is stretched between the tops of the posts. We need to find how far apart the two posts are.
Let’s label the points:
- Point A is the top of the shorter post (8 feet).
- Point B is the top of the taller post (17 feet).
- The wire connects these two points (A and B).
We know the vertical distance between the two posts is the difference in their heights:
[ \text{Vertical Distance} = 17 – 8 = 9 \, \text{feet}. ]
The wire forms the hypotenuse of a right triangle, where:
- One leg is the horizontal distance between the posts (this is what we need to find).
- The other leg is the vertical distance, which we just calculated as 9 feet.
- The hypotenuse is the wire, which is 24 feet long.
Step 2: Applying the Pythagorean Theorem
The Pythagorean Theorem states that for a right triangle:
[
a^2 + b^2 = c^2
]
where (a) and (b) are the legs, and (c) is the hypotenuse. In this case:
- (a) is the horizontal distance between the posts (let’s call this (x)).
- (b) is the vertical distance (9 feet).
- (c) is the length of the wire (24 feet).
Using the Pythagorean theorem:
[
x^2 + 9^2 = 24^2
]
[
x^2 + 81 = 576
]
[
x^2 = 576 – 81
]
[
x^2 = 495
]
[
x = \sqrt{495}
]
[
x \approx 22.27 \, \text{feet}
]
Step 3: Conclusion
The horizontal distance between the two posts, or how far apart they are, is approximately 22.27 feet.
Explanation
The problem uses the Pythagorean theorem to relate the horizontal distance, vertical difference in height, and the wire length (hypotenuse). By applying the theorem, we find the distance between the posts to be about 22.27 feet. The key to solving this was recognizing the right triangle formed by the wire and the difference in heights.