Question

Highway travel A state patrol station is located on a straight north-south freeway. A patrol car leaves the station at 9:00 A.M. heading north with position function s = f(t) that gives its location in miles t hours after 9:00 A.M. (see figure). Assume s is positive when the car is north of the patrol station. <IMAGE>
a. Determine the average velocity of the car during the first 45 minutes of the trip.

Accepted Answer

First, understand that the average velocity is defined as the change in position divided by the change in time. Mathematically, this is expressed as: $$ 	ext{Average Velocity} = \frac{f(t_2) - f(t_1)}{t_2 - t_1} $$, where $$ t_1 $$ and $$ t_2 $$ are the initial and final times, respectively. Convert the time interval from minutes to hours since the position function $$ s = f(t)  is $$given in terms of hours. The first 45 minutes is equivalent to 0.75 hours. Identify the initial time $$ t_1 $$ and the final time $$ t_2 . $$Here, $$ t_1 = 0 $$ hours (9:00 A.M.) and $$ t_2 = 0.75 $$ hours (9:45 A.M.). Evaluate the position function $$ f(t)  at$$ $$ t_1 $$ and $$ t_2  to $$find $$ f(t_1) $$ and $$ f(t_2) . $$These values represent the car's position at the start and end of the 45-minute interval. Substitute $$ f(t_1) $$, $$ f(t_2) $$, $$ t_1 $$, and $$ t_2 $$ into the average velocity formula to calculate the average velocity over the first 45 minutes.

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Key Concepts

Position Function

Average Velocity

Time Conversion