A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 0.8 ft/s, how fast (in rad/s) is the angle (in radians) between the ladder and the ground changing when the bottom of the ladder is 8 ft from the wall? (That is, find the angle's rate of change when the bottom of the ladder is 8 ft from the wall.)On a coordinate plane the horizontal axis is labeled Ground and the vertical axis is labeled Wall. A right triangle is formed by a line of length 10 running down and to the right from the Wall to the Ground. The distance from the origin to the intersection of the line and the Wall is labeled y. The distance from the origin to the intersection of the line and the Ground is labeled x. rad/s

To solve this problem, we can use the relationship between the sides of a right triangle and its angles, which is given by the trigonometric functions. In this case, we can use the cosine function, which is defined as the adjacent side (the distance from the bottom of the ladder to the wall) divided by the hypotenuse (the length of the ladder).

Let's denote the angle between the ladder and the ground as θ, the distance from the bottom of the ladder to the wall as x, and the length of the ladder as L. Then we have:

cos(θ) = x / L

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

-sin(θ) * dθ/dt = dx/dt / L

We are given that dx/dt = 0.8 ft/s (the rate at which the bottom of the ladder is sliding away from the wall), and we want to find dθ/dt (the rate at which the angle is changing). We also know that L = 10 ft, and we can find sin(θ) using the Pythagorean theorem:

sin(θ) = sqrt(L^2 - x^2) / L = sqrt(10^2 - 8^2) / 10 = sqrt(36) / 10 = 0.6

Substituting these values into the equation, we get:

-0.6 * dθ/dt = 0.8 / 10

Solving for dθ/dt, we find:

dθ/dt = -0.8 / (10 * 0.6) = -0.133 rad/s

So the angle between the ladder and the ground is changing at a rate of -0.133 rad/s when the bottom of the ladder is 8 ft from the wall. The negative sign indicates that the angle is decreasing, which makes sense because as the bottom of the ladder slides away from the wall, the ladder gets closer to the ground.