college algebra

Question

A taxi company charges $3.00 for the first mile (or part of a mile) and 60 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.

 if 0 < x ≤ 1

 

    if 1 < x ≤ 1.1

 

    if 1.1 < x ≤ 1.2 

 

    if 1.9 < x ≤ 2

and sketch

Answer 1

Hi Kayla! Great question. So this question has 2 parts to it.

First part is what is the cost for the first mile: thats going to be $3 as a flat fee. So that means whether we are going for 0.5 miles, 0.9, or 1 mile the taxi company charges 3. In piecewise function format, we denote as such:

C(x) = 3 if 0 < x ≤ 1

So that would be our first piecewise function. The second piecewise function relates to every subsequent mile driven. Now, we know the $3 flat fee occurs for the first mile, so that has to be included in this second equation as well. In addition to this, we know it costs 60 cents (or $0.6) for every subsequent mile after the first mile. Now this bold part is important, because we need to create an equation that disregards the first mile, how we would denote that looks like this:

C(x) = 3 + 0.6(x-1)

The 3 is $3 flat fee we have to have in our equation, 0.6 is the rate for every tenth of a mile after, and the (x-1) I just added takes into account miles after the first mile by subtracting the total miles by 1. In piecewise notation we would write it as below:

C(x) = 3 + 0.6(x-1) if 1 < x ≤ 2

Now we want to use this equation to figure out what the subsequent rates are. We would do this by looking into each interval, starting with the second one you wrote:

if 1 < x ≤ 1.1

This "tenth" ends at 1.1. This means we want to figure out what the cost would be at 1.1 miles by plugging this into our second C(x) equation:

C(1.1) = 3 + 0.6(1.1-1)

C(1.1) = 3 + 0.6(0.1)

C(1.1) = 3 + 0.06

C(1.1) = 3.06

So this means that within, and including, the first tenth of a mile after 1 mile (1.1 miles), the cost would be $3.06. In piecewise function notation this would look like:

C(x) = 3.06 if 1 < x ≤ 1.1

We would do the same thing for the second interval you wrote: if 1.1 < x ≤ 1.2 

C(1.2) = 3 + 0.6(1.2-1)

C(1.2) = 3 + 0.6(0.2)

C(1.2) = 3 + 0.12

C(1.2) = 3.12

Similarly the piecewise would look like:

C(x) = 3.06 if 1 < x ≤ 1.1

The last interval will follow the same procedure: if 1.9 < x ≤ 2

C(1.1) = 3 + 0.6(2-1)

C(1.1) = 3 + 0.6(1)

C(1.1) = 3 + 0.6

C(1.1) = 3.6

The final piecewise function would look like:

C(x) = 3.6 if 1.9 < x ≤ 2

Unfortunately I can't graph this equation because I can't upload paper, but you would follow normal graphing procedure with inequalities, open and closed circles.

Please let me know if that helped and if you have any questions!