A 10 feet long ladder leaning against a wall, reaches the wall at a point 8 feet high. By how much distance should the ladder be moved towards the wall so that its top reaches a point at 9.6 feet high?

Initial length of ladder = 10 feet

Initial height reached by the ladder = 8 feet

Desired height after moving ladder = 9.6 feet

When a ladder leans against a wall, it forms a right triangle with the wall and the ground. The length of the ladder is the hypotenuse, the height it reaches on the wall is one leg, and the distance from the wall to the base of the ladder is the other leg.

Using the Pythagorean theorem for both positions:

​$$d_1 = \sqrt{\text{(Length of ladder)}^2 - (\text{Initial height})^2}$$​​

​$$d_2 = \sqrt{\text{(Length of ladder)}^2 - (\text{New height})^2}$$​​

The distance the ladder should be moved is:

\text{Distance Moved} = $$d_1 - d_2$$​​

Let $$d_1$$​  and $$d_2$$​ represent the distances of the ladder's base from the wall in the initial and final positions, respectively.

the initial distance from the wall, $$d_1​$$​:

​$$d_1 = \sqrt{10^2 - 8^2} = \sqrt{100 - 64} = \sqrt{36} = 6 \text{ feet}$$​​

the new distance from the wall, d_2​, when the ladder reaches 9.6 feet:

​$$d_2 = \sqrt{10^2 - 9.6^2} = \sqrt{100 - 92.16} = \sqrt{7.84} \approx 2.8 \text{ feet}$$​​

Now, the distance the ladder should be moved:

Distance Moved = 6 − 2.8 = 3.2 feet

Thus, The ladder should be moved 3.2 feet closer to the wall.