MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A cyclist must travel from point A to point B. Point A and point B are on opposite sides of a rectangular park. The cyclist can ride along a paved road that borders the north side of the park at 18 km/h or cut straight across the grass inside the park at 10 km/h. The park is 0.6 km wide (north–south) and 1.2 km long (east–west). Point A is located at the northwest corner of the park, and point B lies on the south side of the park directly 0.3 km east of the southwest corner.
The cyclist starts at A, rides east along the paved road for y kilometers (where 0≤y≤1.2), leaves the road at point C, and then rides in a straight line across the grass from C to B.
(a) Write an expression for the total travel time T(y) in hours as a function of the distance y traveled on the paved road.
(b) Determine the value of y that yields the shortest travel time. Clearly show algebraic or calculus justification.
(c) Calculate the minimum travel time, to the nearest second, using the optimal value of y. State your answer with appropriate units.