A guy wire is used to stabilize a utility pole. The wire is fastened to the pole 25 feet above the ground, and staked into the ground 4 feet from the base of the pole. (No stand-offs are used, the wire goes directly from the pole to the ground stake.) Someone recommends that more stability could be achieved by moving the ground stake out to 6 feet from the pole.

a. How long is the guy wire with the present arrangement?
b. How much longer would the guy wire have to be with the proposed arrangement?

a) So you're looking at a Right Angled Triangle (RAT) with the two short sides being 4 and 25. What does Pythagoras say about the other side, which must be the hypotenuse?

b) This is the same question, except with 6 on the shortest side instead of 4.

twenty feet

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To find the length of the guy wire, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

a. With the present arrangement:
We have a right triangle formed by the pole, the ground, and the guy wire. The guy wire is the hypotenuse, the distance from the base of the pole to the ground stake is one side, and the distance from the pole to the ground (the height) is the other side.

Using the Pythagorean theorem, we can calculate the length of the guy wire:
Length of guy wire = √(side₁^2 + side₂^2)

In this case:
side₁ = height = 25 feet
side₂ = distance from base to ground stake = 4 feet

Length of guy wire = √(25^2 + 4^2)
Length of guy wire = √(625 + 16)
Length of guy wire = √641
Length of guy wire ≈ 25.31 feet

Therefore, the guy wire with the present arrangement is approximately 25.31 feet long.

b. With the proposed arrangement:
If the ground stake is moved out to 6 feet from the pole, we now have a new right triangle with an updated distance from base to ground stake.

Using the same formula as before, we can calculate the new length of the guy wire:
Length of guy wire = √(side₁^2 + side₂^2)

In this case:
side₁ = height = 25 feet
side₂ = new distance from base to ground stake = 6 feet

Length of guy wire = √(25^2 + 6^2)
Length of guy wire = √(625 + 36)
Length of guy wire = √661
Length of guy wire ≈ 25.74 feet

Therefore, with the proposed arrangement, the guy wire would be approximately 25.74 feet long.

To summarize:
a. With the present arrangement, the guy wire is approximately 25.31 feet long.
b. With the proposed arrangement, the guy wire would be approximately 25.74 feet long.