Problem 59 - Algebra 2

Question

BUSINESS Reliable Rentals rents cars for \(\$ 12.95\) per day plus 15\(\not{c}\) per mile. Luis Romero works for a company that limits expenses for car rentals to \(\$ 90\) per day. How many miles can Mr. Romero drive each day?

Step-by-Step Solution

Answer

Mr. Romero can drive up to 513 miles each day.

Step 1: Understanding the Problem

We need to find out the maximum number of miles Luis Romero can drive in a day without exceeding the $90 expense limit set by his company for renting a car.

Step 2: Setting Up the Equation

The cost to rent a car is comprised of a daily base fee and a mileage charge. The equation is: \( 12.95 + 0.15m \leq 90 \), where \( m \) represents the number of miles driven.

Step 3: Isolating the Variable

Subtract \(12.95\) from both sides to isolate the mileage charge: \( 0.15m \leq 77.05 \).

Step 4: Solving for the Maximum Miles

Divide both sides by \(0.15\) to find \(m\): \( m \leq \frac{77.05}{0.15} \approx 513.67 \).

Step 5: Interpreting the Result

Since Mr. Romero cannot drive a fraction of a mile, round \(513.67\) down to the nearest whole number. Hence, he can drive a maximum of 513 miles.

Key Concepts

Algebraic Expressions

Algebraic expressions are like a mathematical language used to represent real-world situations using numbers and symbols. In our exercise, the daily cost for the car rental is expressed as an algebraic expression. This expression includes a fixed term, \(12.95\), which is the daily base fee for renting the car, and a variable term, \(0.15m\), which represents the cost per mile driven where \(m\) is the variable showing the number of miles driven.
This way, algebraic expressions help us turn word problems into mathematical equations. They consist of terms, which are parts of an expression that are added together. Terms may include numbers, known values (coefficients), and variables representing unknown values.
For instance, in the expression \(12.95 + 0.15m\), \(12.95\) is a constant term, \(0.15\) is a coefficient, and \(m\) is a variable. Using algebraic expressions makes it easier to solve complex real-world problems by manipulating these expressions and ultimately finding the values of unknowns.

Problem Solving with Mathematics

Problem solving with mathematics involves taking a real-world problem and converting it into a mathematical model that can be worked through step by step. The car rental scenario we examined is a great example of breaking down a problem.
First, we need to understand the problem by identifying what is known and what needs to be found. Here, the trick is knowing that the total daily cost must not exceed \(\$ 90\).
Then, we set up and solve an algebraic inequality: \(12.95 + 0.15m \leq 90\). By solving this inequality, we isolate the variable to find how many miles can be driven within the budget.
The steps generally involve simplifying expressions, applying mathematical operations such as subtraction and division, and ultimately solving for the variable. This structured approach not only helps in solving the problem at hand but also in developing critical thinking skills that can be applied to a variety of situations.

Real-world Applications of Algebra

Using algebra to solve real-world problems allows us to find practical solutions to everyday issues, like budgeting a car rental in our exercise. Algebra helps us make sense of complex scenarios involving costs and limits, transforming them into manageable mathematical problems.
In business, for example, knowing how to calculate and forecast expenses is crucial. By using algebraic equations, companies can create budgets, set limits, and plan better. The car rental example shows this by calculating how many miles can be driven without surpassing a budget.
Furthermore, the same reasoning applies to other scenarios like planning travel distances, managing personal finances, or even engineering problems where resource constraints are essential. Algebra equips us with the tools to set up and calculate with mathematical precision, leading to informed decision-making in various aspects of life.