Problem 52 - Calculus Volume 1

Question

[T] 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) \(C(t) = 20 + 10.25t\), b) \(583.75\), c) \(40.27\) hours.

Step 1: Understand the Problem

We need to first identify the flat fee and the hourly charge. The problem states a flat fee of \(20\) dollars as a flat fee for renting the car, and \(10.25t\), where \(t\) is the number of hours the car is rented. Thus, the function that models this situation is: \[ C(t) = 20 + 10.25t \]

Step 3: Calculate Total Cost for Given Time

Now, we will calculate the total cost of renting the car for a specific time period. First, convert 2 days into hours: \(2\) days = \(2 \times 24 = 48\) hours. Add the 7 hours: \(48 + 7 = 55\) hours.Next, plug \(t = 55\) into the function \(C(t) = 20 + 10.25t\):\[ C(55) = 20 + 10.25 \times 55 \]

Step 4: Perform Calculation

Compute the total cost:\[ C(55) = 20 + 10.25 imes 55 \]\[ C(55) = 20 + 563.75 \]\[ C(55) = 583.75 \]Therefore, the total cost to rent the car for 2 days and 7 hours is \(583.75\) dollars.

Step 5: Solve for Rental Duration given Total Cost

Now, find \(t\) when the total cost is \(432.73\). Using the equation:\[ 432.73 = 20 + 10.25t \]Subtract the flat fee of \(20\) from both sides:\[ 432.73 - 20 = 10.25t \]\[ 412.73 = 10.25t \]Solve for \(t\):\[ t = \frac{412.73}{10.25} \]

Step 6: Final Calculation

Now perform the division:\[ t = \frac{412.73}{10.25} \approx 40.27 \]Thus, the car was rented for approximately \(40.27\) hours.

Key Concepts

Cost Calculation

Cost calculation is an essential part of understanding how expenses accumulate over time, especially in real-world situations like car rentals. The problem provides a clear example: a rental car company charges a flat fee and an hourly rate. Understanding where each cost comes from allows for accurate budgeting and expense tracking.

In this problem, we are given a flat fee of \(20 and an hourly charge of \)10.25. To calculate the total cost for any rental duration, we must consider both the fixed and variable parts of this equation. The flat fee remains constant, no matter how many hours the car is rented. Meanwhile, the hourly charge accumulates with each hour the car is used. Thus, the total cost formula can be modeled as:

• The flat fee: 20
• The hourly rate multiplied by the number of hours: 10.25 * t

This leads us to the function: \( C(t) = 20 + 10.25t \). Once we understand these components, solving for specific rental costs or durations becomes a simple task of plugging in numbers.

Function Modeling

Function modeling involves creating a mathematical representation of a real-world situation. In this context, we model the car rental charges based on time. Understanding how to construct a function is crucial for analyzing different scenarios and making informed decisions.

The primary goal of function modeling is to express a relationship between quantities. Here, we relate the duration of the car rental (\(t\)) to its total cost (\(C\)). We start by identifying the constants in our problem, such as the flat fee and hourly charge. Then, we translate these values into a linear function.

• The fixed part of costs is represented as: \(20\)
• The variable part is expressed as: \(10.25t\)

Combining these, the function \( C(t) = 20 + 10.25t \) models how total cost changes with time. This predictable relationship dictates every calculation we perform in problem-solving.

Problem-solving Steps

Effective problem-solving involves a structured approach to breaking down the situation and finding solutions. This problem requires sequential understanding and application of concepts.

Firstly, we define the problem and write the function \( C(t) = 20 + 10.25t \). Next, for part (b), calculating the cost for a specific rental duration means substituting the total hours into the function. Converting days into hours is crucial here: 2 days equal 48 hours, added to 7 hours gives 55 hours. Plugging 55 into our function, we find:

• Cost for 55 hours: \( 20 + 10.25 \times 55 = 583.75 \)

For part (c), solving for \(t\) when the bill is $432.73 requires rearranging the function: \( 432.73 = 20 + 10.25t \). Subtraction and division help isolate \(t\), showing the need for precise calculations.

• Finding \(t\): \( t = \frac{412.73}{10.25} \approx 40.27 \)

Following these steps ensures clarity and accuracy in determining costs and rental durations.