Question

A 5-meter-long ladder is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall at a rate of 2 m/sec. Determine how fast its height on the wall is decreasing when the foot of the ladder is 4 meters away from the wall?

Answer 1

The height of the ladder on the wall is decreasing at a rate of -8/3 m/s when the foot of the ladder is 4 meters away from the wall.

Step-by-step explanation:

To determine how fast the height of the ladder on the wall is decreasing, we can use similar triangles. Let the height of the ladder on the wall be represented by 'h' and the distance of the foot of the ladder from the wall be represented by 'x'. Since the ladder is leaning against the wall, we can form a right triangle with the ladder as the hypotenuse.

Using the Pythagorean theorem, we have:

h² + x² = 5²

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

2h(dh/dt) + 2x(dx/dt) = 0

Substituting the given values of x = 4 m and dx/dt = 2 m/s:

2h(dh/dt) + 2(4)(2) = 0

Plugging in h = √(5² - 4²) = 3 m, we can solve for dh/dt:

2(3)(dh/dt) + 16 = 0

6(dh/dt) = -16

dh/dt = -16/6 = -8/3 m/s