We need to determine the length of a guywire that extends from the top of a 30ft pole to a point on the ground 20ft from the base of the pole. This can be modeled as a right triangle, where the pole is one leg, the distance on the ground is the other leg, and the guywire is the hypotenuse. We need to find the length of the hypotenuse.

1. Problem Description

We need to determine the length of a guywire that extends from the top of a 30ft pole to a point on the ground 20ft from the base of the pole. This can be modeled as a right triangle, where the pole is one leg, the distance on the ground is the other leg, and the guywire is the hypotenuse. We need to find the length of the hypotenuse.

2. Solution Steps

We can use the Pythagorean theorem to solve for the length of the guywire. The Pythagorean theorem states that for a right triangle with legs of length $a$ and $b$, and a hypotenuse of length $c$, the following equation holds:

$a^2 + b^2 = c^2$

In this problem:

$a = 30$ ft (height of the pole)

$b = 20$ ft (distance on the ground from the base of the pole)

$c$ = length of the guywire (which we want to find)

Plugging in the values for $a$ and $b$ into the Pythagorean theorem, we get:

$30^2 + 20^2 = c^2$

$900 + 400 = c^2$

$1300 = c^2$

Now we take the square root of both sides:

$c = \sqrt{1300}$

$c = \sqrt{100 \cdot 13}$

$c = \sqrt{100} \cdot \sqrt{13}$

$c = 10\sqrt{13}$

$c \approx 36.0555$

So the length of the guywire is approximately 36.06 ft.

3. Final Answer

The length of the guywire must be approximately $10\sqrt{13}$ ft, or approximately 36.06 ft.