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Chapter 12: Problem 2

Use the following information to answer the next three exercises. A vacation resort rents SCUBA equipment to certified divers. The resort charges an up- front fee of \(\$ 25\) and another fee of \(\$ 12.50\) an hour. Find the equation that expresses the total fee in terms of the number of hours the equipment is rented.

Short Answer

Expert verified
The equation is \( C = 25 + 12.5h \).

Step by step solution

01

Identify the Fixed Fee

The resort charges an upfront fee of \( \$ 25 \). This is a fixed cost that does not change regardless of the hours the equipment is rented.
02

Identify the Variable Fee

The resort charges \( \$ 12.50 \) per hour for renting the equipment, which means this cost depends on the number of hours the equipment is used.
03

Define the Variable

Let \( h \) represent the number of hours the equipment is rented. This variable will be used to express the variable fee in terms of hours.
04

Construct the Equation

The total cost \( C \) is the sum of the fixed fee and the variable fee. This can be represented as \( C = 25 + 12.5h \), where \( 25 \) is the upfront fee and \( 12.5h \) is the fee per hour multiplied by the number of hours \( h \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fixed Fee
In many real-world scenarios, the concept of a "Fixed Fee" is integral when considering costs related to services such as rentals. A fixed fee is a one-time charge that remains constant regardless of how much or how little a service is used. This is often referred to as a "base fee" or "service fee."

For example, in the context of a vacation resort that rents SCUBA equipment, there's an upfront charge of $25. This amount is the fixed fee charged to the customer before they even start using the equipment.
  • The fixed fee does not vary with time.
  • No matter the duration the equipment is rented, the fixed fee remains $25.
  • It ensures the service provider covers basic administration costs.
Understanding the fixed fee is crucial as it sets the baseline for any transaction, ensuring that both parties have a clear and consistent starting point for costs.
Variable Fee
A "Variable Fee" differs from a fixed fee because it changes with the usage of a service. This type of fee is calculated based on the quantity or time of service utilized.

In our SCUBA equipment rental scenario, the resort charges $12.50 per hour. This is the variable fee because it depends directly on the number of hours rented:
  • The more hours the equipment is rented, the higher the total cost due to this variable fee.
  • If the equipment is used for 0 hours, no variable fee is incurred.
This structure allows flexibility for users, as they pay precisely for the duration the service is utilized. Each additional hour of use increases the cost by $12.50, making it essential to plan accordingly depending on budget constraints or rental needs.
Cost Function
The "Cost Function" integrates both fixed and variable fees to provide a complete understanding of the total expenditure based on the usage of a service.

Mathematically, it is expressed as:
  • The cost function for the SCUBA equipment rental is defined as: \[ C = 25 + 12.5h \].
Here:
  • \( C \) represents the total cost.
  • The number 25 is the fixed fee.
  • \( 12.5h \) is the variable fee, where \( h \) is the number of rental hours.
This equation allows customers to calculate their anticipated costs based on the length of time they wish to rent the equipment. For instance, if a diver rents the equipment for 3 hours, the total cost would be:\[ C = 25 + 12.5(3) = 62.5 \]Thus, the cost function provides a practical way to budget and manage expenses related to rental services.

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