A tree is supported by a wire anchored in the ground 11 feet from its base. The wire is 4 feet longer than the height that it reaches on the tree. Find the length of the wire. 
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Part 1
The wire is 
  
enter your response here feet long.​ (Type an​ integer, proper fraction or mixed​ number.)

We can solve this problem by applying the Pythagorean theorem, which states that in a right triangle:

\[
a^2 + b^2 = c^2
\]

Where:
- \( a \) is the distance from the tree's base to the point where the wire is anchored in the ground (11 feet),
- \( b \) is the height of the tree reached by the wire,
- \( c \) is the length of the wire.

According to the problem, the length of the wire \( c \) is 4 feet longer than the height \( b \) it reaches on the tree. This can be expressed as:

\[
c = b + 4
\]

Now, substitute this into the Pythagorean theorem:

\[
11^2 + b^2 = (b + 4)^2
\]

Expanding the equation:

\[
121 + b^2 = b^2 + 8b + 16
\]

Simplify by canceling \( b^2 \) on both sides:

\[
121 = 8b + 16
\]

Subtract 16 from both sides:

\[
105 = 8b
\]

Solve for \( b \):

\[
b = \frac{105}{8} = 13.125 \, \text{feet}
\]

Now that we know \( b = 13.125 \), we can find \( c \), the length of the wire:

\[
c = b + 4 = 13.125 + 4 = 17.125 \, \text{feet}
\]

So, the length of the wire is:

\[
\boxed{17.125} \, \text{feet}
\]

Would you like further details or clarification? 

Here are 5 related questions for further practice:
1. What is the height that the wire reaches on the tree?
2. How would the solution change if the wire were 6 feet longer than the height?
3. Can you verify the solution using the Pythagorean theorem directly with these values?
4. How would the solution change if the wire were anchored 15 feet from the base of the tree?
5. How can you derive a general formula for any such problem where the wire is \( x \) feet longer than the height?

**Tip:** Always double-check your equation setups, especially when incorporating conditions like "the wire is longer than the height."

A tree is supported by a wire anchored in the ground 11 feet from its base. The wire is 4 feet longer than the height that it reaches on the tree. Find the length of the wire.

This problem involves using the Pythagorean theorem to find the length of a wire supporting a tree. The wire is anchored 11 feet from the base and is 4 feet longer than the height it reaches on the tree. Learn how to solve this step by step.

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