MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A 13-ft ladder rests against a vertical wall. The bottom of the ladder slides away from the wall along a horizontal floor. At the instant when the bottom is 5 ft from the wall, the bottom moves horizontally at a rate of 3 ft/s. Let θ denote the angle between the ladder and the floor.
(a) Calculate the rate at which the top of the ladder is descending at that instant. Include appropriate units.
(b) Determine dtdθ at the same instant and state whether θ is increasing or decreasing. Provide justification.
(c) The ladder is equipped with a paint roller attached 2 ft below its top. Find the vertical speed of the paint roller at the instant described above.
(d) Interpret the result of part (b) in practical terms for someone standing near the ladder base. Refer explicitly to the numeric value and sign of dtdθ. No additional calculation is required.