A bicyclist must travel from point A on the south bank of a straight river to point B…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A bicyclist must travel from point A on the south bank of a straight river to point B on the north bank. Point A and point B are 600 m apart in the east-west direction, and the river is 400 m wide. From A the cyclist may ride east along the south bank at 5 m/s to a launch point P, 0 ≤ x ≤ 600, where x is the eastward distance from A to P. From P the cyclist rows a small boat directly to B at 2 m/s (relative to still water). Ignore river current.
Assume the cyclist rides in a straight line along the bank to P and then rows straight to B.
(a) Write an expression for the total travel time T(x) as a function of x.
(b) Show that T(x) has exactly one critical point in the open interval (0, 600).
(c) Determine the value of x that minimizes the travel time and find the corresponding minimum time, both to the nearest tenth of a second.
(d) Verify that your answer satisfies the conditions for an absolute minimum on the closed interval [0, 600], and explain the practical meaning of the endpoints x = 0 and x = 600.