Question

a telephone pole has a wire attached to its top that is anchored to the ground. the distance from the bottom of the pole to the anchor point is 42 feet less than the height of the pole. if the wire is to be 6 feet longer than the height of the pole, what is the height of the pole?

Explanation:

Define variables and expressions

Using the Pythagorean Theorem Applications knowledge point
Let \(h\) be the height of the pole in feet.
The anchor point distance is \(h - 42\) feet.
The length of the wire is \(h + 6\) feet.

Set up the equation

Using the Pythagorean Theorem Applications knowledge point

$$ h^2 + (h - 42)^2 = (h + 6)^2 $$

Expand and simplify the quadratic equation

Using the Solving Quadratic Equations knowledge point

$$
LATEXBLOCK0
$$

Solve the quadratic equation

Using the Solving Quadratic Equations knowledge point

$$
LATEXBLOCK1
$$

Verify the physical constraints

Using the Pythagorean Theorem Applications knowledge point
Since the anchor distance \(h - 42\) must be positive:

$$ h - 42 > 0 \implies h > 42 $$

Thus, \(h = 72\) is the only valid height.

Answer:

A telephone pole has a wire attached to its top that is anchored to the ground. The distance from the bottom of the pole to the anchor point is 42 feet less than the height of the pole. If the wire is to be 6 feet longer than the height of the pole, what is the height of the pole?

<blank>72</blank> feet