A ladder rests against a vertical pole. The foot of the ladder is sliding away

from the pole along horizontal ground. Find the inclination of the ladder to
the horizontal at the instant when the top of the ladder is moving 3 times as
fast as the foot of the ladder.

To find the inclination of the ladder to the horizontal, we can use trigonometry. Let's break down the problem and follow these steps:

Step 1: Identify the relevant quantities in the problem.
- Let's call the distance from the foot of the ladder to the pole "x" (in meters).
- Let's call the distance from the top of the ladder to the pole "y" (in meters).
- Let's call the angle between the ladder and the ground "θ" (in degrees).

Step 2: Understand the relationship between the quantities.
- We are given that the top of the ladder is moving 3 times as fast as the foot of the ladder. This means that the rate of change of "y" with respect to time is three times the rate of change of "x" with respect to time:
dy/dt = 3dx/dt

Step 3: Relate the quantities using trigonometry.
- We can use the Pythagorean theorem to relate the distances "x" and "y". According to the theorem, the square of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides:
x^2 + y^2 = L^2, where L is the length of the ladder.

Step 4: Differentiate the equation to involve rates of change.
- Differentiate both sides of the equation with respect to time (t):
d(x^2)/dt + d(y^2)/dt = d(L^2)/dt
2x(dx/dt) + 2y(dy/dt) = 0, since the length of the ladder (L) is constant.

Step 5: Substitute the given relationship between dx/dt and dy/dt.
- Since dy/dt = 3dx/dt, we can substitute it into the equation:
2x(dx/dt) + 2y(3dx/dt) = 0
2x(dx/dt) + 6xy(dx/dt) = 0

Step 6: Simplify the equation and solve for dx/dt.
- Factor out dx/dt:
dx/dt(2x + 6xy) = 0
dx/dt = 0 (since 2x + 6xy ≠ 0 for any x and y)

Step 7: Determine the values of x and y at the given scenario.
- We are told that the top of the ladder is moving 3 times as fast as the foot of the ladder. This means dx/dt ≠ 0. Therefore, the only solution is 2x + 6xy = 0.

Step 8: Solve for the inclination angle θ.
- We can use the tangent function to find θ:
tanθ = y/x
θ = atan(y/x)

By following these steps, we can find the inclination of the ladder to the horizontal at the given instant.