a 15 ft ladder is palced against a wall such that its base is 9 ft away from the wall. if the base of the ladder is moved to 6 ft away from the wall, by about how much will the top of the ladder move up the wall?

at 9 ft, you have a 9-12-15 triangle.

at 6 ft, the ladder reaches to height h, where

6^2 + h^2 = 15^2
h = 13.75 ft

so, the ladder has moved up 1.75 ft

To find out how much the top of the ladder will move up the wall, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

In this scenario, the ladder, the distance it moves horizontally (base), and the distance it moves vertically (height) form a right triangle. So, we can use the Pythagorean theorem to solve for the height.

Let's denote the initial distance between the base of the ladder and the wall as "a" (9 ft) and the height of the ladder on the wall as "b" (unknown).

According to the Pythagorean theorem, we have:

a^2 + b^2 = c^2,

where "c" is the length of the ladder (15 ft).

Plugging in the values, we get:

9^2 + b^2 = 15^2,

81 + b^2 = 225.

To solve for "b," we need to isolate the variable on one side of the equation. Subtracting 81 from both sides gives us:

b^2 = 225 - 81,

b^2 = 144.

Finally, taking the square root of both sides, we find:

b = sqrt(144),

b = 12 ft.

So, initially, the top of the ladder is 12 ft up the wall.

Now, let's move the base of the ladder to 6 ft away from the wall. To find out how much the top of the ladder will move up the wall, we can apply the same steps as before.

Using the Pythagorean theorem again, we have:

6^2 + b^2 = 15^2,

36 + b^2 = 225.

Subtracting 36 from both sides gives us:

b^2 = 225 - 36,

b^2 = 189.

Taking the square root of both sides, we find:

b = sqrt(189),

b ≈ 13.75 ft.

Therefore, when the base of the ladder is moved to 6 ft away from the wall, the top of the ladder will move up approximately 13.75 ft.