a 65 foot ladder is leaning against a well. Its lower end is 25 feet away from the wall. How much farther away will it be if upper end is moved down 8 feet?

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Explanation:

Check out the diagrams below.

We'll start with the left diagram (marked "before") which is a right triangle with the horizontal leg of 25 feet and hypotenuse 65 feet.

Use the pythagorean theorem to find the vertical side x.

a^2 + b^2 = c^2

25^2 + x^2 = 65^2

625 + x^2 = 4225

x^2 = 4225 - 625

x^2 = 3600

x = sqrt(3600)

x = 60

The top of the ladder is 60 feet high when placed against the wall in this configuration.

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If the upper end is moved down 8 feet, then x-8 = 60-8 = 52 feet is the new height of the ladder. Refer to the "after" in the diagram below.

Like earlier, we'll use the pythagorean theorem to find the missing side.

a^2 + b^2 = c^2

y^2 + 52^2 = 65^2

y^2 + 2704 = 4225

y^2 = 4225 - 2704

y^2 = 1521

y = sqrt(1521)

y = 39

The horizontal distance from the ladder base to the wall is now 39 feet.

Earlier it was 25 feet, so it has increased by 39-25 = 14 feet.

the awnser is visionary