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Function modeling involves creating a mathematical representation of a situation or problem. In this case, we're using a linear function to depict the relationship between time (hours) and cost for renting a car. The linear function captures two essential components: a fixed starting fee and a variable cost dependent on time.

• The fixed component here is the flat fee of \(\\(20\), which is constant regardless of how long the car is rented.
• The variable component is the hourly rate of \(\\)10.25\), which depends on the number of hours the car is rented.

By understanding these two parts, we construct the function: \[ C(t) = 20 + 10.25t \] This equation clearly shows how total cost \(C\) changes linearly with time \(t\). Modeling functions this way allows us to predict costs for varying rental periods, making complex real-world situations easier to handle.

Algebraic equations are crucial for solving real-world problems by representing unknowns. In our rental car scenario, the key equation is the one linking cost \(C\) to time \(t\):
\[ C = 20 + 10.25t \] This linear equation has an easy-to-calculate structure. It helps gauge cost when the rental time is known or find rental time when the total cost is given.
To solve these equations:

• You determine your target (cost or time) and isolate the variable in the equation.
• In situations like part a, you might know the hours and need the cost, easily solved by substituting \(t\) into the function.
• For parts like c, where the cost is fixed and you need to find \(t\), you rearrange and solve the equation \(C = 20 + 10.25t\) using basic algebraic manipulation: isolate \(t\) by performing reverse operations (subtraction and division).

Mastering these steps simplifies the interpretation and application of algebraic equations.

Effective problem-solving in algebra involves a systematic approach. Here’s how you tackle problems such as this rental car cost scenario:
1. **Define the function and understand variables**: Recognize fixed and variable aspects. Define the function reflecting these attributes. 2. **Break down time units carefully**: Remember hour and day conversions. Convert larger units to the smallest unit (hours here) to simplify calculations. 3. **Substitute known values**: Plug in values for quick answers. For costs related to known rental times, directly substitute into the linear function. 4. **Solve for unknowns**: When the outcome (like total cost) is given, rearrange and solve the function for unknown factors (hours, in this case). 5. **Verify your results**: Once you find a solution, check calculations by substituting back into the function. This step ensures no arithmetic errors.
Implementing these orderly steps helps with minimizing mistakes and provides a clear roadmap to understand and solve problems efficiently.