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# Minutes and Miles

Suppose you plan to make a trip of L miles and you begin driving at a speed of v0 miles per minute. You have a choice of gaining speed in two ways:

• (A) gain speed at a rate of 0.1 miles per minute every minute
• (B) gain speed at a rate of 0.1 miles per minute every mile

The speed gain is continuous in either case, meaning that in (A) you undergo acceleration of 0.1 in the familiar form of mi/min/min, and in (B) your acceleration is at the rate of 0.1 (mi/min)/mile, in a continuous fashion based on miles.

One can ask specific questions such as:

• What are the (A)-travel time and (B)-travel time when v0 = 1.5 and L = 50?
• What are the (A)-travel time and (B)-travel time when v0 = 0.5 and L = 50?
• What are the (A)-travel time and (B)-travel time when v0 = 0.5 and L = 15?

Problem 1042: What is the minimum initial speed v0 such that (B) gives the faster travel time for all values of L?

Extra Credit: Find the region in the (v0, L) plane for which timeA < timeB.

Source: Book by Wm. Briggs, Ants, Bikes, and Clocks, Problem Solving for Undergraduates (SIAM), p. 101.