Problem 45 - Algebra A Combined Function

Question

Hertz Car Rental charges a daily rate of \(\$ 39\) plus \(\$ 0.20\) per mile for a certain car. Suppose that you rent that car for a day and your bill (before taxes) is \(\$ 95\). How many miles did you drive?

Step-by-Step Solution

Answer

The car was driven for 280 miles.

Step 1: Understand the Problem

We need to find out how many miles a car was driven when the cost to rent it was $95 for one day, with a daily rental charge of $39 plus $0.20 per mile driven.

Step 2: Set Up the Equation

Let \( x \) represent the number of miles driven. The total cost of renting the car is given by the expression \( 39 + 0.20x \). We know this total is $95.

Step 3: Write the Equation

We have the equation for the cost as \( 39 + 0.20x = 95 \), where \( 39 \) is the daily rate charge and \( 0.20x \) is the cost for the miles driven.

Step 4: Solve for x (Miles Driven)

Subtract 39 from both sides of the equation to isolate the term with \( x \): \[ 39 + 0.20x - 39 = 95 - 39 \] which simplifies to: \[ 0.20x = 56 \]. Next, divide both sides by 0.20 to solve for \( x \): \[ x = \frac{56}{0.20} \].

Step 5: Calculate

Calculate \( \frac{56}{0.20} \): \[ x = 280 \].Thus, the car was driven for 280 miles.

Key Concepts

linear equations

Linear equations are fundamental tools in algebra used to describe relationships between variables. In this case, the variables are the cost of renting a car and the number of miles driven. A linear equation typically looks like this: \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents a variable we want to solve for.

Here's how they apply to our car rental problem:

• The daily rate of the car is a fixed cost, represented by \( 39 \).
• The mileage rate is a variable cost, changing with the number of miles driven, represented by \( 0.20\times x \) (where \( x \) is the number of miles).
• Our goal is to find \( x \) when the total cost (\( 39 + 0.20x \)) equals \( 95 \).

By setting up and solving the linear equation, you uncover the relationship between costs and mileage as a straight line where changes in one variable directly affect the other. This linkage is the essence of linear equations in real-world problems.

problem solving

Problem-solving in mathematics involves understanding the problem, strategizing a plan, carrying out calculations, and reflecting on the solution for accuracy. For the car rental scenario, this process is vital.

Initially, understanding the problem means recognizing that every aspect of the rental cost can be described with a simple equation. It involves identifying known information like daily fixed and variable costs.

• Strategizing involves converting these pieces into an equation, \( 39 + 0.20x = 95 \), which consolidates the problem's components into a solvable format.
• The solution involves straightforward algebraic manipulation by isolating \( x \), using basic operations like subtraction and division.
• Finally, reviewing the outcome ensures that our calculations reflect the real-world scenario appropriately. In this case, driving 280 miles causes the cost to reach the given \( 95 \) dollars, aligning perfectly with the problem's requirements.

Every step of this problem-solving process builds alongside mathematical intuition, improving proficiency with similar algebraic challenges in the future.

cost analysis

Cost analysis breaks down components of expenses to understand the total outlay. This technique helps determine the economic rationale behind financial decisions, like renting a car.

In our scenario, cost analysis reveals the breakdown of rental charges:

• The car has a base rental charge per day, a fixed cost of \( \\(39 \).
• The cost per mile driven, a variable cost, amounts to \( \\)0.20 \) for each mile.
• To find how these costs accumulate, formulate an expression for total cost \( 39 + 0.20x \), and equate it to our total of \( \$95 \).

By decomposing the cost elements into fixed and variable components, we gain insights into how different factors, like distance driven, impact overall expenses. This awareness allows for informed decision-making, predicting how changes in mileage affect the rental's cost, and ensuring budget-conscious travel plans.