Greenwood press

130
STEP FUNCTIONS

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fore the phone-call function can also be written as 
p = 0.12ceil(t). Any phone
call between 3.01 and 4.00 minutes will result in the same charge, or any phone
call between 4.01 and 5.00 minutes will result in the same charge, and so on. The
figure below illustrates the price of a phone call as a function of its time.
Other rates that use discrete values for pricing can often be modeled with step
functions. The price to mail a package is dependent on its mass according to a step
function. If the cost to deliver a letter is 34 cents for the first ounce and 23 cents
for each additional ounce, then the function 
p = 0.23
⌈m − 1⌉ + 0.34 describes
the total price in dollars, 
p, as a function of the mass in ounces, m. This equation
is slightly different than the one for the price of a phone call, because there is a
different rate for the first ounce. The 
⌈m − 1⌉ portion of the equation accounts for
the additional price of any mass above one ounce. You can determine this rela-
tionship in the equation because any value of 
m between 0 and 1 will cause the
quantity
⌈m − 1⌉ to equal 0, meaning that nothing additional to 34 cents will be
added to the cost of postage for mail that is between 0 and 1 ounces.
Consulting and repair rates are often represented by step functions. A visit to
an attorney’s office might be $100 for making an appointment, and then an addi-
tional $150 per hour, or fraction thereof. That means that an hour-and-a-half
appointment would be equivalent to a $400 fee—$100 for showing up and $300
for two hours of work. Sometimes rates are divided into smaller increments of
time, such as with automobile repair. Some auto shops may charge $80 per hour,
and make charges to the next one-half hour. That means that a car that has been
repaired for an hour and 13 minutes will be charged for 1.5 hours of labor, or
$120. As a step function, the repair cost in dollars, 
r, in terms of the number of
hours of labor, 
h, is represented by the equation r = 40⌈2h⌉. This equation needs
to consider the number of half-hour intervals, since the overall charge is rounded
to the nearest half-hour. The 
2h in the equation describes the number of half-
hours of labor, and the 40 represents the half-hour rate of $40.
The cost of a taxicab ride also relates to a step function in terms of the dis-
tance traveled. Often there is an initial amount charged for getting in the cab, like
$2.70, and then an additional fee, like $0.30, for every block or fraction of a
block traveled. In this case, a 9.3-block cab ride would cost
2.70 + 0.30⌈9.3⌉, or
$5.70. Notice that the distance traveled would be equivalent to 10 blocks, since
there is not a specific fee for 0.3 blocks. In fact, in most cases involving fees or
costs paid by the consumer, rates are usually rounded up with a ceiling function.