A ladder is needed to reach the top of a second floor window, A distance of 20 feet above the (level) ground. According to the directions on the ladder, the distance between the bottom of a leaning ladder and the base of the (vertical) wall it is leaning up against should be 30% of the length of the ladder. Find the length of the ladder that will just to reach the top of the second-floor window when positioned as the directions describe.

If the ladder has length z, then

(.3z)^2+20^2 = z^2
z ≈ 21 ft

To find the length of the ladder needed to reach the top of the second-floor window, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the ladder forms the hypotenuse, while the distance between the bottom of the ladder and the base of the wall is one of the other sides. Let's call this distance x.

According to the directions, the distance between the bottom of the ladder and the base of the wall should be 30% of the length of the ladder. In other words, x = 0.3l, where l is the length of the ladder.

The height of the second-floor window is the other side of the right triangle, which is 20 feet.

Now, we can apply the Pythagorean theorem:

l^2 = x^2 + (20^2)

Substituting x = 0.3l:

l^2 = (0.3l)^2 + (20^2)

Simplifying, we have:

l^2 = 0.09l^2 + 400

Subtracting 0.09l^2 from both sides:

0.91l^2 = 400

Dividing both sides by 0.91:

l^2 = 400 / 0.91

l^2 ≈ 439.56

Taking the square root of both sides:

l ≈ √439.56

l ≈ 20.94

Therefore, the length of the ladder needed to reach the top of the second-floor window, when positioned as the directions describe, is approximately 20.94 feet.