A 24 foot ladder is leaning against a building. Let x be the distance between the bottom of the ladder abd the building and let theta be the angle between the ladder and the ground. Suppose the bottom of the ladder is sliding away from the wall at a rate of 2 ft/sec. When x=6, how fast is theta (in Radians/sec) changing?
cosØ = x/24
-sinØ dØ/dt = (1/24)dx/dt
when x = 6, opposite side is y
y^2 + 6^2 = 24^2
y = √540 and sinØ = √540/24
dØ/dt = -1/(24sinØ)dx/dt
= (-1/24)(24/√540)(2) = -2/√540 rad/sec
simplify as needed
To determine how fast theta is changing, we need to use related rates.
Let's denote the length of the ladder (24 feet) as L, the distance between the base of the ladder and the building (x), and the angle formed by the ladder and the ground (theta). We are given that dx/dt (the rate at which x is changing) is 2 ft/sec.
Using trigonometry, we have the relationship: L*cos(theta) = x.
Differentiating both sides of this equation with respect to t (time), we get:
-d/dt(L*cos(theta)) = d/dt(x).
Since L is a constant (24 ft), the derivative of L*cos(theta) with respect to t is:
-d/dt(L*cos(theta)) = -L*sin(theta)*(d(theta)/dt).
So, the equation becomes:
-L*sin(theta)*(d(theta)/dt) = dx/dt.
Plugging in the given values when x = 6 and dx/dt = 2, we have:
-24*sin(theta)*(d(theta)/dt) = 2.
Now, we need to solve for d(theta)/dt, which represents how fast theta is changing:
d(theta)/dt = 2 / (-24*sin(theta)).
Since we want the rate of change in radians per second, we need to convert the angle theta from degrees to radians. The conversion factor is: 1 radian = π/180 degrees.
Therefore, we have:
d(theta)/dt = 2 / (-24*sin(theta)) * (π/180) rad/sec.
Hence, when x = 6, the speed at which theta is changing (in radians/sec) can be found using the formula:
d(theta)/dt = 2 / (-24*sin(theta)) * (π/180) rad/sec.