Question

a ladder 25 feet long is leaning against a barn. it is sliding down the barn at a rate of 3 ft/sec. how fast, in ft/sec, is the foot of the ladder moving away from the wall at the moment that the foot of the ladder is 7 feet from the wall?

Answer 1

The foot of the ladder is moving away from the wall at a rate of 3 ft/sec.

Step-by-step explanation:

To find the rate at which the foot of the ladder is moving away from the wall, we can use the concept of related rates. Let's denote the distance between the foot of the ladder and the wall as x, and the distance between the top of the ladder and the ground as y.

Since the ladder is sliding down the barn at a rate of 3 ft/sec, we can express the relationship between x and y using the Pythagorean theorem: x^2 + y^2 = 25^2.

Now, let's differentiate both sides of the equation with respect to time (t):

2x(dx/dt) + 2y(dy/dt) = 0. Since x is given as 7 feet, and we want to find dy/dt (the rate at which the foot of the ladder is moving away from the wall), we can substitute these values into the equation and solve for dy/dt.

Plugging in the values:

2(7)(dx/dt) + 2y(dy/dt) = 0.

Since dx/dt = 0 (the distance from the foot of the ladder to the ground is not changing), we have:

14(dy/dt) = -2(7)(3).

Simplifying the equation:

14(dy/dt) = -42.

Dividing both sides of the equation by 14:

dy/dt = -3 ft/sec.

Therefore, the foot of the ladder is moving away from the wall at a rate of 3 ft/sec.