A 12-foot ladder rests against a vertical wall on a level floor
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A 12-foot ladder rests against a vertical wall on a level floor. The bottom of the ladder is pulled straight away from the wall, remaining in contact with the floor at all times.
Throughout the motion, let x(t) be the horizontal distance (in feet) from the wall to the foot of the ladder and let y(t) be the height (in feet) of the top of the ladder above the floor. At one instant the foot of the ladder is 5 ft from the wall and moving horizontally at 2 ft/s away from the wall.
(a) Give a clear verbal justification for the equation that relates x(t) and y(t) during the motion.
(b) At the instant described, indicate the sign (positive, negative, or zero) of dtdy and justify your answer without computing its numerical value.
(c) Decide whether the magnitude of dtdy is less than, equal to, or greater than the magnitude of dtdx at this instant. Provide a convincing argument based on the geometry of the situation rather than an explicit calculation.
(d) Using appropriate reasoning, explain how your conclusions in parts (b) and (c) are consistent with the physical behavior of the ladder.