A 20-ft ladder rests against a vertical wall on a level surface
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A 20-ft ladder rests against a vertical wall on a level surface. The foot of the ladder is being pulled away from the wall at a constant rate of 1 ft/s.
(a) At the instant when the foot of the ladder is 12 ft from the wall, determine whether the top of the ladder is moving upward or downward. Explain your reasoning in terms of appropriate rate signs and units.
(b) Without performing a numerical calculation, explain how the magnitude of the vertical speed of the ladder’s top changes as the foot continues to slide farther from the wall. Justify your answer by reference to the geometric relationship among the three sides of the right triangle formed by the ladder, the wall, and the floor.
(c) Will there ever be an instant at which the magnitude of the horizontal speed of the foot equals the magnitude of the vertical speed of the top? Provide a justification that includes algebraic or graphical reasoning.