Question

Airport Parking Valuepark charge \(\$ 1.10\) per hour or fraction of an hour for airport parking. The maximum charge per day is \(\$ 7.25\) (a) Write a formula that gives the charge for \(x\) hours with \(0 \leq x \leq 24 .\) (Hint: See Exercise \(52 . )\) (b) Graph the function in part (a). At what values of \(x\) is it continuous?

Short answer

The formula for parking charges is \(f(x) = 1.10 * \lceil x \rceil\) for \(0 \leq x < 6.59\) and \(f(x) = 7.25\) for \(6.59 \leq x \leq 24\). From the graph, the function is discontinuous at \(x = 6.59\) and continuous at all other points between 0 and 24.

Step-by-step solution

Understand the Charging Mechanism and Write a Formula

The charge for any parking hour or fraction of an hour is $1.10. This means that even a fraction of an hour, say 0.1 hour, is also charged as a full hour ($1.10). Therefore, the number of hours is always rounded up to the nearest whole number. Thus, it can be represented as \(\lceil x \rceil\), where \(\lceil x \rceil\) is the ceiling function that rounds a number up to the nearest integer. \n\nThe charge for a number of hours is $1.10 times the number of hours, until the maximum charge of $7.25 per day is reached. Hence, the function can be represented as two cases in mathematical form: \(f(x) = 1.10 * \lceil x \rceil\) for \(0 \leq x < 6.59\) and \(f(x) = 7.25\) for \(6.59 \leq x \leq 24\)

Graph the Function

The function \(f(x)\) has two parts. From \(0 \leq x < 6.59\), it's an increasing function because the cost increases with the number of hours. This part of the function will look like steps rising from the origin to reach a height of $7.25 at \(x = 6.59\). \n\nFrom \(6.59 \leq x \leq 24\), the cost is flat at $7.25 because the cost doesn't increase beyond this point. This part of the function will be a flat line at a height of $7.25 from \(x = 6.59\) to \(x = 24\).

Analyze Continuity

Looking at the graph, it can be observed that this function is discontinuous at \(x = 6.59\). This is because at \(x = 6.59\), the function jumps from $7.15 to $7.25. At all other points, \(f(x)\) is continuous because there are no jumps or breaks in the graph.