Mathematics

Problem

A shipping company charges for packaging based on the weight

w

(in pounds) of a parcel. The cost

C

(in dollars) is modeled by

C(w)=\begin{cases} p, & 0\le w < 5,\\ 1.2w+3, & w \ge 5, \end{cases}

where

p

is a fixed surcharge for parcels under

5\,\text{lb}

. a. Part 1: (a) Write an equation involving

p

that must be satisfied for

C

to be continuous at the threshold weight

w=5

. b. Part 2: (b) Solve for

p

so that the cost function is continuous for all

w\ge 0

. c. Part 3: (c) In practical terms, describe why continuity of the cost function at

w=5

is important from the perspective of customer billing.