Final answer:
The rate at which the top of a ladder slides down as the bottom is pulled away is calculated using related rates in calculus. By applying the Pythagorean theorem and differentiating with respect to time, we find that the top slides down at 4 m/s when the bottom is 4 m from the wall.
Step-by-step explanation:
The student asked about the rate at which the top of a ladder slides down a wall as the bottom is pulled away horizontally at a specific rate when the ladder is 4m from the base of the post. This is a related rates problem in calculus, a part of mathematics that involves differential equations.
To solve this problem, we can apply the Pythagorean theorem to the triangle formed by the ladder, the ground, and the wall. Let x be the distance from the base of the wall to the ladder's bottom, and y be the height of the ladder on the wall. The length of the ladder is the hypotenuse and is constant at 5 m. The problem gives dx/dt = 3 m/s and asks for dy/dt when x = 4 m.
The Pythagorean theorem states that x2 + y2 = 52. Differentiating both sides with respect to t gives us 2x (dx/dt) + 2y (dy/dt) = 0. We can solve for dy/dt knowing that x = 4 m and dx/dt = 3 m/s. Plugging in the values, we find that when x = 4 m, y = 3 m. So, 2(4)(3) + 2(3)(dy/dt) = 0. Solving for dy/dt gives us dy/dt = -4 m/s indicating that the top of the ladder slides down at 4 m/s when the bottom is 4 m from the base.