A snowmobile starts from rest at point O on the south edge of a frozen lake
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A snowmobile starts from rest at point O on the south edge of a frozen lake. The clinic lies 500 m north and 300 m east of O. The driver follows this plan:
• Accelerate in a straight line along the southern shoreline with constant acceleration 2 m/s² for a distance x m (0 ≤ x ≤ 300), reaching speed v=4x m/s at the launch point P.
• Immediately turn and travel in a straight line across the ice from P to the clinic at the constant speed v attained at P.
Assume the turn takes negligible time and that the path on the lake is straight.
(a) Express the total travel time T(x) in terms of x.
(b) Compute dxdT and determine the critical point(s) of T on the interval [0, 300].
(c) Show that exactly one critical point yields an absolute minimum travel time, and find the corresponding value of x and minimum time to the nearest tenth of a second.
(d) Provide a physical interpretation of why accelerating longer than the optimal distance increases total travel time despite achieving a higher speed.