The top of a ladder slides down a vertical wall at a rate of 0.375 m/s0.375 m/s . At the moment when the bottom of the ladder is 5 m5 m from the wall, it slides away from the wall at a rate of 0.9 m/s0.9 m/s . How long is the ladder?

This problem can be solved using the Pythagorean theorem, which 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 lengths of the other two sides.

Let's denote:

• L as the length of the ladder
• x as the distance from the bottom of the ladder to the wall
• y as the height from the top of the ladder to the ground

From the Pythagorean theorem, we have:

L² = x² + y²

Differentiating both sides with respect to time (t), we get:

2L * (dL/dt) = 2x * (dx/dt) + 2y * (dy/dt)

Given that the top of the ladder slides down the wall at a rate of 0.375 m/s, we have dy/dt = -0.375 m/s (the negative sign indicates that y is decreasing).

Given that the bottom of the ladder slides away from the wall at a rate of 0.9 m/s, we have dx/dt = 0.9 m/s.

At the moment when the bottom of the ladder is 5 m from the wall, we have x = 5 m.

We can find y using the Pythagorean theorem. Since the ladder hasn't changed its length, we have:

y = sqrt(L² - x²) = sqrt(L² - 5²)

Substituting these values into the differentiated equation, we get:

2L * (dL/dt) = 250.9 + 2sqrt(L² - 5²)(-0.375)

Since the ladder doesn't change its length over time, we have dL/dt = 0, which simplifies the equation to:

0 = 9 - 0.75*sqrt(L² - 25)

Solving this equation for L gives us the length of the ladder.

0.75*sqrt(L² - 25) = 9

sqrt(L² - 25) = 9 / 0.75

L² - 25 = (9 / 0.75)²

L² = 25 + (9 / 0.75)²

L = sqrt(25 + (9 / 0.75)²)

Calculating the above expression will give us the length of the ladder.