Question

A rental car company charges \(\$ 20\) for one day, allowing up to 200 miles. For each additional 100 miles, or any fraction thereof, the company charges \(\$ 18\). Sketch a graph of the cost for renting a car for one day as a function of the miles driven. Discuss the continuity of this function.

Short answer

The cost function is a step function with jumps at each additional 100-mile increment beyond 200 miles.

Step-by-step solution

Understand the Problem

The problem involves two components: a base rate for renting a car for one day and an additional charge based on the distance traveled beyond a certain mileage.

Define the Function

The cost function, \( C(x) \), represents the total cost based on miles driven, \( x \). The base cost is \( \\(20 \) for up to 200 miles. Beyond 200 miles, an additional \( \\)18 \) is charged for every 100 miles (or fraction thereof).

Determine Cost Intervals

For \( 0 \leq x \leq 200 \), the cost \( C(x) = 20 \). For \( x > 200 \), the cost becomes \( C(x) = 20 + 18 \times \lceil (x-200)/100 \rceil \), where \( \lceil \cdot \rceil \) denotes the ceiling function, rounding up to the nearest whole number.

Sketch the Graph

Plot \( C(x) \) on the graph. The function is constant at \( \\(20 \) for \( 0 \leq x \leq 200 \). For each subsequent 100-mile increment beyond 200 miles, the cost increases by \( \\)18 \), creating step-like increases at 300, 400, and so on.

Discuss Continuity

The function has jump discontinuities at every multiple of 100 miles beyond 200 miles (e.g., 300, 400, etc.), where the cost suddenly increases by \( \$18 \), making the graph a step function after 200 miles.

Key concepts

Understanding the Step Function

Step functions are unique mathematical concepts that are particularly useful when dealing with situations that involve sudden jumps or changes. In the context of the car rental company's cost function, the step function appears once you surpass the 200-mile limit. The cost remains fixed at $20 for any mileage up to 200 miles. However, as soon as you surpass the 200-mile mark, the cost increases by $18 for every additional full or partial 100 miles.
This jump creates a step-like pattern on the graph. Hence, for mileages like 201 to 300, the cost would jump to $38, creating a distinct horizontal segment. Once it reaches 301 miles, the cost jumps again, illustrating another step and continuing this staircase pattern.

The Role of Discontinuities

Discontinuities in a function occur when there is an abrupt change or 'jump' in values. This is very apparent in the cost function for the car rental. The graph of the function experiences jump discontinuities at 200 miles and every subsequent multiple of 100 miles thereafter (e.g., 300, 400, etc.).
Discontinuities occur because the cost has to jump by $18 every time another 100 miles—full or partial—are driven beyond the initial 200 miles. These points make it impossible for the function to be continuous, as there's no smooth transition. Instead, the function displays sudden, finite gaps at these key mileage points.

Breaking Down Mileage Charges

Mileage charges play a crucial role in determining the overall cost when renting a vehicle under these rules. The base charge of $20 covers up to 200 miles. This is straightforward and involves no mileage charge beyond the base fee within this range.
However, beyond these 200 miles, for each increment of 100 miles or even part of an increment, an additional charge of $18 is added. Thus, if you drive 201 miles, you will still pay an extra $18, as the policy rounds up to the nearest 100-mile segment using the ceiling function. This creates additional charges rapidly if more miles are driven.

Utilizing the Ceiling Function

The ceiling function is a mathematical tool used to round numbers up to the nearest whole number. In this scenario, it is applied to determine the additional charges for mileage over 200 miles. The formula in use is:\[ C(x) = 20 + 18 \times \left\lceil \frac{x - 200}{100} \right\rceil \]This formula ensures that any fraction of additional 100 miles results in a complete charge of $18. For example, driving 201 miles requires you to pay as if you drove 300 miles, both rounded up, emphasizing the ceiling function's role.

• This function rounds up any decimal or fraction to the next integer.
• It ensures that even a small increase over the benchmark mileage results in an additional charge.

By utilizing this function, the company ensures they maximize the mileage charges efficiently for every customer driving beyond the base limit.