Question

A taxi company charges $4.00 for the first mile (or part of a mile) and 40 cents for each succeeding tenth of a mile (or part). Express the cost C (in dollars) of a ride as a piecewise defined function of the distance x traveled (in miles) for 0 < x ≤ 2.

Answer 1

Here's one way of defining the function.

Explanation:

A piecewise-defined function consists of two or more equations, each of which is valid for some interval.

Your function will start at $4.00 for the first mile and then increase in $0.40 steps for each 0.1 mi to a maximum of $8.00.

[[x]] means the greatest integer less than or equal to x.

In this function [[x]] = 1.

`C(x) =\begin{cases}4.00& \quad 0 \leq x \leq 1 \\4.00 + 0.40[[x]] & \quad 1 < x \leq 1.1\\4.40 + 0.40[[x]] & \quad 1.1 < x \leq 1.2\\4.80 + 0.40[[x]] & \quad 1.2 < x \leq 1.3\\5.20 + 0.40[[x]]& \quad 1.3 < x \leq 1.4\\5.60 + 0.40[[x]] & \quad 1.4 < x \leq 1.5\\6.00 + 0.40[[x]] & \quad 1.5 < x \leq 1.6\\6.40 + 0.40[[x]] & \quad 1.6 < x \leq 1.7\\6.80 + 0.40[[x]] & \quad 1.7 < x \leq 1.8\\7.20 + 0.40[[x]] & \quad 1.8 < x \leq 1.9\\7.60 + 0.40[[x]] & \quad 1.9 < x \leq 2\\\end{cases}`

A graph of your function is shown below.

Answer 2

See below

Explanation:

`4.00, 0<x\leq 1 \\4.40, 1<x\leq 1.1 \\4.80, 1.1<x\leq 1.2 \\5.20, 1.2<x\leq 1.3 \\5.60, 1.3<x\leq 1.4 \\6.00, 1.4 <x\leq 1.5 \\6.40, 1.5 <x\leq 1.6 \\6.80, 1.6 <x\leq 1.7 \\7.20, 1.7<x\leq 1.8 \\7.60, 1.8 <x\leq 1.9 \\8.00, 1.9 <x\leq 2\\`

Since it charges $4.00 for the first mile or part of the first mile. If the taxi has traveled 0.5 miles it will charge 4.00, so any travel between 0 and 1 mile (including 1 mile) costs 4.00. and each tenth of a mile or a part (0.1 mile) costs 40 cents, meaning 0.04 miles will cost 40 cents too.

For example is a travel is 1.24 miles,

First mile = 4.00

so you still have to pay the additional 0.24 miles,

0.2 miles costs 40*2 cents = 80 cents

and the additional 0.04 miles are considered as another tenth, another 40 cents.

Total = 5.20 cents