Hey cywhite,
I am assuming that you are asking for the form of equation for the following conditions:
t<=3 hours, Cost= $5 per hour
t> 3 hours, Cost=$4 per hour
Cost(t)=5*t when t<=3
Cost(t)=15+4*(t-3) when t>3
To combine the two, you need to use step function or heavy side function:
Cost(t)=5*t*step(3-t)+(15+4*(t-3))*step(t-3).
When the argument of step function is positive, it tends to 1, and when the argument becomes negative, it tends to zero.
Depending on how the step function is defined at t=3 in your textbook when the argument becomes zero, you may need to slightly modify the above equation.
Sometimes, step(0)=1/2, in such cases, the earlier equation is fine. but if step(0)=1, you may want something like the below:
Cost=5*t*step(3-t)+(15+4*(t-3))*step(t-3\(\displaystyle ^+\)), where \(\displaystyle 3^+\) is value infinitesimally greater than 3.
Equation for:
$5 for each hour, after 3 hours every additional time is $4.
