Question

A company that operates for 18 hours a day has introduced a new daily wage system that
incorporates variable hourly rate to encourage longer shifts. The company pays $10 for first
hour or any part of the first hour. Afterwards, company pays additional 2% per hour for each
hour or part of each additional hour. Write a piece-wise function that shows the salary as a
function of hours. Also sketch a graph of this function.

Answer 1

Let $t$ represent the hours and let $f(t)$ be the salary as function of hours $t$.

For the first hour or any part of the first hour;

$f(t)=10~~~0<t\leq1$

For the additional hours. That is $1<t\leq18$

$f(t)=10+(2\%\times10)(t-1)\\ f(t)=10+0.2t-0.2\\ f(t)=9.8+0.2t$

Hence the piece-wise function representing the salary is\\

$f(t)=\begin{cases} 10 ~~~~~~~~~~~~~~~~~~~~0<t\leq 1\\ 9.8+0.2t ~~~~~~~~1<t\leq18 \end{cases}$

The graph is as follows