Final answer:
The tip of the ladder is slipping at a rate of 0.22 feet per second when it is 5.2 feet away from the house. The angle is changing at a rate of approximately -0.066 radians per second when the ladder is 5.2 feet away from the house.
Step-by-step explanation:
To find how fast the tip of the ladder is slipping along the side of the house, we can use the concept of similar triangles. Let's denote the distance between the bottom of the ladder and the house as x and the distance between the tip of the ladder and the house as y. We have the following relationship:
x/y = length of ladder from bottom to tip / length of ladder from bottom to house
Plugging in the values, we get: x/5.2 = 18/18, which simplifies to: x = 5.2
Since we know x is changing at a rate of 0.22 feet per second, we can calculate how fast y is changing by differentiating the equation with respect to time: dy/dt = (dy/dx) * (dx/dt).
dy/dt = (18/18) * 0.22 = 0.22
So, the tip of the ladder is slipping along the side of the house at a rate of 0.22 feet per second when the ladder is 5.2 feet away from the house.
To find how fast the angle is changing, we can use trigonometry. Let's denote the angle between the ladder and the ground as θ. We have the following relationship:
cos(θ) = y / length of ladder from bottom to tip
Plugging in the values, we get: cos(θ) = 5.2 / 18, which simplifies to: θ ≈ 70.61°
Now, we can differentiate the equation with respect to time to find how fast the angle is changing: dθ/dt = -sin(θ) * (dy/dt) / (length of ladder from bottom to tip)
Plugging in the values, we get: dθ/dt = -sin(70.61) * 0.22 / 18 ≈ -0.066 radians per second
Therefore, the angle is changing at a rate of approximately -0.066 radians per second when the ladder is 5.2 feet away from the house.