The top of a ladder slides down a vertical wall at a rate of 0.15 m/s. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s. How long is the ladder?

Solution:

Given, the top of a ladder slides down a vertical wall at a rate of 0.15 m/s.

When the bottom of the ladder is 3m from the wall, it slides away from the wall at a rate of 0.2 m/s.

We have to find the length of the ladder.

Let y be the height up the wall to the top of the ladder.

Let x be the distance between the bottom of the wall and the bottom of the ladder.

Let L be the length of the ladder.

Given, y is decreasing at 0.15 m/s

dy/dt = - 0.15 m/s

Given, x is increasing at 0.2 m/s when x = 3m.

dx/dt = 0.2 m/s when x = 3.

Using Pythagoras theorem,

L² = x² + y²

On differentiating,

0 = 2x(dx/dt) + 2y(dy/dt)

Dividing by 2 on both sides,

0 = x(dx/dt) + y(dy/dt)

Putting the values of dx/dt, x and dy/dt in the above expression,

(3)(0.2) + y(-0.15) = 0

0.6 - 0.15y = 0

0.15y = 0.6

So, y = 0.6/0.15

y = 4 m

When x = 3 m, y= 4 m

So, L² = (3)² + (4)²

= 9 + 16

= 25

L² = 25

Taking square root,

L = √25

L = 5 m

Therefore, the length of the ladder is 5 m.

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The top of a ladder slides down a vertical wall at a rate of 0.15 m/s. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s. How long is the ladder?

Summary:

The top of a ladder slides down a vertical wall at a rate of 0.15 m/s. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s. The ladder is 5 m long.