A 13 foot ladder leaning against a wall makes an angle of degree radians with the ground. The base of the ladder is pulled away from the wall at a rate of 2ft/sec. How fast is degree changing when the base of the ladder is 5 feet way.

if the base is x from the wall,

cosθ = x/13
-sinθ dθ/dt = 1/13 dx/dt
when x=5, sinθ = 12/13

-12/13 dθ/dt = 1/13 (2)
dθ/dt = -1/6

note the "-" sign: the angle decreases as the ladder slips down.

5, 12, 13 right triangle

call angle ladder to ground T
cos T = x/13 where x is base of ladder to base of wall

-sin T dT/dt = (1/13) dx/dt

at x = 5, t = 0, dx/dt = 5
cos T = 5/13 so T = 67.4 degrees or 1.18 radians

sin T = sin 67.4 = 12/13 = .923

-.923 dT/dt = (1/13)(2)
so
dT/dt = -.167 radians/second
times 180/pi = -9.55 degrees/second

To find how fast the angle degree is changing, we need to use trigonometric functions and differentiate with respect to time. Let's break down the problem and set up the equation.

Let θ represent the angle the ladder makes with the ground, and let x represent the distance between the base of the ladder and the wall.

We are given that the length of the ladder is 13 feet, and it is leaning against the wall, so we can form a right triangle.

Using trigonometry, we know that:

sin(θ) = opposite/hypotenuse = x/13

To differentiate both sides with respect to time t, we get:

d/dt(sin(θ)) = d/dt(x/13)

Next, we need to use the chain rule. The derivative of sin(θ) with respect to t is cos(θ) times dθ/dt, and since dθ/dt represents how fast the angle is changing, which is what we are looking for, we can rewrite this as:

cos(θ) * dθ/dt = (1/13) * dx/dt

Now, we need to find dx/dt, which represents the rate at which the base of the ladder is pulled away from the wall. We are given that dx/dt is 2 ft/sec. Therefore, we have:

cos(θ) * dθ/dt = (1/13) * 2

To find the value of cos(θ), we use the Pythagorean theorem. Since we have a right triangle, we can find the length of the opposite side (x) and the hypotenuse (13) using the equation:

x^2 + 13^2 = 13^2

x^2 = 13^2 - 13^2

x^2 = 0

x = 0

This tells us that when the base of the ladder is 5 feet away from the wall, the triangle collapses into a degenerate triangle with zero area. At this point, the ladder is laying flat on the ground.

Therefore, when x = 0, the angle θ is undefined, and we can't find the value or rate of change of θ.