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The Parking Lot


Alignments to Content Standards: F-IF.A.1

Task

A parking lot charges $\$0.50$ for each half hour or fraction thereof, up to a daily maximum of $\$10.00$. Let $C(t)$ be the cost in dollars of parking for $t$ minutes.

  1. Complete the table below.

    $t$ (minutes)$C(t)$ (dollars)
    0
    $15$
    $20$
    $35$
    $75$
    $125$
  2. Sketch a graph of $C$ for $0 \leq t \leq 480$.
  3. Is $C$ a function of $t$? Explain your reasoning.
  4. Is $t$ a function of $C$? Explain your reasoning.

IM Commentary

The purpose of this task is to investigate the meaning of the definition of function in a real-world context where the question of whether there is more than one output for a given input arises naturally. In more advanced courses this task could be used to investigate the question of whether a function has an inverse.

Solution

  1. $t$ (minutes)$C(t)$ (dollars)
    00
    $15$$0.50$
    $20$$0.50$
    $35$$1.00$
    $75$$1.50$
    $125$$2.50$
    As a sample calculation, we note that $125$ minutes is two full hours (four half-hours) and part of another half hour. Since the ticketing scheme rounds up to the nearest half-hour, we have to pay for five half-hours, at a total cost of \$2.50.
  2. Redone_parking_lot_b8361c7ad2f974944824d6f4cb058dca
  3. Yes, $C$ is a function of $t$ because for a given parking time of $t$ minutes there is exactly one charge.
  4. No, $t$ is not a function of $C$ because there are values of $C$ that have many values of $t$ associated with them. For example if you end up paying $\$0.50$ then you could have parked for any period of time up to half an hour, that is, when $C = 0.50$ then $t$ can have any value in the range $0 < t \le 30$. So the "input" $C = 0.50$ yields more than one output, which is not allowed for a function.

Cam says:

over 2 years

I have no objection to additional clarity, though I'm not entirely sure it was unclear in the first place (and not sure that adding clarity adds meaning, either). There's also the point that the language currently employed in the task is closer to what actual signage on such parking lots would contain. Maybe we could get some more opinions to weigh in on this.

Jambo says:

almost 3 years

Suggest greater clarity at the expense of brevity in the question. That is, clarify that the cost is $.50 even if only a minute of a half hour is used. See how a second sentence helps add meaning?

David Jabon says:

about 5 years

Suggestion for a small improvement: give the units in the table headers: t (hours) and C(t) (dollars).

Cam says:

about 5 years

Good idea. Thanks.