Question

Math IIIB Performance Task: Piecewise-Defined Functions

Question Set 2

A parking garage charges the following amounts for cars parked in the garage:
For the first hour that a car is parked in the garage, there is no charge.
After the first hour, for the next two hours that a car is parked in the garage, there is a $5 charge.
After the third hour, the garage charges $2 for each additional hour that the car is parked in the garage. If a car is parked in the garage for a fraction of an hour, the garage will charge that fraction of the additional hourly rate.


Use the information provided about the parking garage to answer each of the following questions. Refer to the responses you submitted for Question Set 1 for support.


Sketch a graph to model the total cost of parking in the garage over the domain [0, 12]. (10 points)




Write the piecewise-defined function for the total cost of parking in the garage. That is, state the function C(x), where x is the number of hours a car is parked in the garage. (10 points)









The parking garage plans to change the structure of how much it charges for cars that park in it. A graph of the new total cost is shown below. Describe in detail the new structure. (10 points)

Answer 1

9514 1404 393

Step-by-step explanation:

a) see the attachment for a graph (red curve)

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b) The piecewise defined function can be ...

`\displaystyle C(x)=\begin{cases}0&0\le x\le 1\\5&1<x\le 3\\2x-1&3<x\le12\end{cases}`

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c) The new cost function is ...

`\displaystyle C(x)=\begin{cases}10&0\le x\le5\\x+5&5<x\end{cases}`

The charge is a flat $10 for any period up to 5 hours, then $1 per additional hour after that, prorated for partial hours. There is apparently no upper limit on parking time as there was with the original function. (The graph shown goes beyond 15 hours; the domain of the original C(x) is 0 to 12 hours.) For reference, the new cost structure is shown in blue in the attachment.

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Additional comment

Under the new structure, costs are significantly higher for short-term parking, and lower once the term exceeds 6 hours.