A wire is attached to the top of a pole. The wire is 4 ft longer than the pole, and the distance

step 1 Okay, so it says the wire is four feet longer than the pole. So if the pole measures X, we know

A wire is attached to the top of a pole. The wire is 4 ft...

A wire is attached to the top of a pole. The wire is $4 \mathrm{ft}$ longer than the pole, and the distance from the wire on the ground to the bottom of the pole is $4 \mathrm{ft}$ less than the height of the pole. Find the length of the wire and the height of the pole.

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Key Concepts

Pythagorean Theorem

The problem involves a right triangle formed by the pole, the ground distance from the pole, and the wire as the hypotenuse. The Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, is used to relate these three quantities.

Algebraic Modeling

The problem requires setting up relationships between the unknown quantities (the height of the pole, the length of the wire, and the horizontal distance). Using algebraic expressions for these relationships—such as one length being a fixed amount larger or smaller than another—allows us to form equations that can be solved systematically.

Quadratic Equation Solving

Once the equation from the Pythagorean theorem is established with the substitutions from the modeled relationships, it typically leads to a quadratic equation. Solving this quadratic equation using methods like factoring or the quadratic formula provides the numerical values of the unknowns.

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