A ladder 20 feet long leans up against a house.
The bottom of the ladder starts to slip away from the house at 0.17 feet per second.
How fast is the tip of the ladder along the side of the house slipping when the ladder is 7.2 feet away from the house? (Round to 3 decimal places.)
ft/sec
Consider the angle the bottom of the ladder makes with the ground.
How fast is the angle changing (in radians) when the ladder is 7.2 feet away from the house?
x^2 + y^2 = 20^2
x dx/dt + y dy/dt = 0
tanθ = y/x
sec^2θ dθ/dt = (x dy/dt - y dx/dt)/x^2
or
cosθ = x/20
-sinθ dθ/dt = 1/20 dx/dt
To solve this problem, we can use trigonometry and related rates. 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 ground as y.
First, let's consider the relationship between the lengths of the ladder, the height of the house, and the distance between the bottom of the ladder and the house. We can use the Pythagorean theorem:
x^2 + y^2 = 20^2
Differentiating this equation implicitly with respect to time:
2x(dx/dt) + 2y(dy/dt) = 0
Since we want to find the rate at which the tip of the ladder is slipping along the side of the house (dy/dt), we need to find dx/dt and substitute it into the equation.
Given that the bottom of the ladder is slipping away from the house at a rate of 0.17 feet per second (dx/dt = 0.17 ft/s), we can substitute this value into the equation:
2x(0.17) + 2y(dy/dt) = 0
Simplifying the equation, we get:
0.34x + 2y(dy/dt) = 0
Now, we need to determine the values of x and y when the ladder is 7.2 feet away from the house. From the given information, we have x = 7.2 ft and we can solve for y using the Pythagorean theorem:
y = sqrt(20^2 - x^2)
= sqrt(20^2 - 7.2^2)
= sqrt(400 - 51.84)
= sqrt(348.16)
≈ 18.641 ft
Substituting these values into the equation:
0.34(7.2) + 2(18.641)(dy/dt) = 0
Simplifying further:
2(18.641)(dy/dt) = -0.34(7.2)
dy/dt = (-0.34 * 7.2) / (2 * 18.641)
dy/dt ≈ -0.082 ft/s
Therefore, the tip of the ladder along the side of the house is slipping at a rate of approximately -0.082 ft/s when the ladder is 7.2 feet away from the house.
To find the rate at which the angle is changing, we can use the trigonometric relation between x, y, and the angle:
cos(angle) = x / 20
Differentiating this equation implicitly with respect to time, we get:
-sin(angle) * (d(angle)/dt) = (dx/dt) / 20
Substituting the given values:
-sin(angle) * (d(angle)/dt) = 0.17 / 20
d(angle)/dt ≈ -0.0085 radians/s
Therefore, the angle is changing at a rate of approximately -0.0085 radians per second when the ladder is 7.2 feet away from the house.