A car travels along a straight

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A car travels along a straight, flat road for

0\le t\le12\,\text{min}

with velocity (in m/min)

v(t)=\begin{cases} k(t-3)^{2}, & 0\le t\le 3,\\ 18, & 3<t\le6,\\ 18-m(t-6), & 6<t\le12. \end{cases}

The constants

k

and

m

are positive. The car is at the origin at

t=0

and is

180\,\text{m}

farther down the road at

t=12\,\text{min}

. (a) Show that the velocity is continuous on the entire interval

0\le t\le12\,\text{min}

without imposing any additional conditions on

k

m

. (b) Compute the car’s position function

s(t)

on each sub-interval in terms of

k

and

m

, using

s(0)=0

as the initial condition. (c) Use the information

s(12)=180

to derive a single equation relating

k

and

m

. (d) Additional testing reveals that the car’s acceleration is continuous at

t=6\,\text{min}

. Use this fact to obtain a second independent equation involving

k

and

m

. (e) Solve the system obtained in parts (c) and (d) to find numerical values for

k

and

m

, then determine the maximum distance from the origin attained by the car during the 12-minute trip.