Problem 52 - Calculus Volume 1

Question

A rental car company rents cars for a flat fee of $$\$ 20$$ and an hourly charge of $$\$ 10.25 .$$ Therefore, the total cost \(C\) to rent a car is a function of the hours \(t\) the car is rented plus the flat fee. a. Write the formula for the function that models this situation. b. Find the total cost to rent a car for 2 days and 7 hours. c. Determine how long the car was rented if the bill is $$\$ 432.73 .$$

Step-by-Step Solution

Answer

a. The formula is \(C(t) = 20 + 10.25t\). b. The total cost is \$583.75. c. The car was rented for approximately 40.26 hours.

Step 1: Define the Function Formula

To model this situation, we need to write a function that represents the total cost of renting a car. The cost consists of a flat fee of \\(20 plus \\)10.25 per hour rented. The formula for the cost \(C\) as a function of time \(t\) in hours is: \[ C(t) = 20 + 10.25t \]

Step 2: Convert Days to Hours

To find the total cost for 2 days and 7 hours, we first convert the days into hours. There are 24 hours in a day, so 2 days is \(2 \times 24 = 48\) hours. Adding 7 more hours gives us a total of \(48 + 7 = 55\) hours.

Step 3: Calculate Total Cost for a Given Time

Using the function from Step 1, substitute \(t = 55\) hours into the formula \(C(t) = 20 + 10.25t\) to find the total cost: \[ C(55) = 20 + 10.25 \times 55 = 20 + 563.75 = 583.75 \] Hence, the total cost for renting the car for 2 days and 7 hours is \$583.75.

Step 4: Find the Duration from the Total Cost

To determine how long the car was rented if the total cost was \$432.73, we solve the equation \(C(t) = 432.73\). Using the formula from Step 1: \[ 432.73 = 20 + 10.25t \] Subtract 20 from both sides to isolate the term with \(t\): \[ 412.73 = 10.25t \] Divide both sides by 10.25 to solve for \(t\): \[ t = \frac{412.73}{10.25} = 40.26 \] Thus, the car was rented for approximately 40.26 hours.

Key Concepts

Function modeling

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

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.

Problem-solving steps

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.