Problem 59 - College Algebra with Corequisite Support

Question

For the following exercises, use this scenario: The cost of renting a car is $$\$ 45 /$$ wk plus $$\$ 0.25 / \mathrm{mi}$$ traveled during that week. An equation to represent the cost would be \(y=45+.25 x,\) where \(x\) is the number of miles traveled. Suppose you have a maximum of $$\$ 100$$to spend for the car rental. What would be the maximum number of miles you could travel?

Step-by-Step Solution

Answer

You can travel up to 220 miles.

Step 1: Write Down the Budget Constraint

The car rental budget for the week is $100. Therefore, we can represent the cost constraint with the inequality equation: \(y \leq 100\).

Step 2: Use the Cost Equation

The equation for the total car rental cost is \(y = 45 + 0.25x\), where \(x\) is the number of miles traveled. We'll use this equation to find the maximum miles under the budget constraint.

Step 3: Substitute the Cost Equation into the Constraint

Substitute the cost equation into the budget constraint: \(45 + 0.25x \leq 100\). This replaces \(y\) in the inequality with an expression in terms of \(x\).

Step 4: Solve for Number of Miles 'x'

Subtract 45 from both sides to isolate the term with \(x\):\[0.25x \leq 55\]Next, divide both sides by 0.25 to solve for \(x\):\[x \leq \frac{55}{0.25} = 220\].

Step 5: Interpret the Solution

The inequality \(x \leq 220\) implies that you can travel a maximum of 220 miles without exceeding your budget of $100.

Key Concepts

Linear Inequalities

Linear inequalities are similar to linear equations, but instead of having an equal sign, they use inequality symbols like \(<, \leq, >, \geq\). In the context of solving real-world problems, linear inequalities help express conditions of limits or constraints, such as budgets.
For example, if you only have $100 to rent a car, you could represent this constraint by the inequality \(y \leq 100\). This reads as "the total cost \(y\) is less than or equal to 100 dollars."
This inequality is then used in combination with other equations to determine possible solutions that meet this constraint.

Cost Equation

A cost equation is a mathematical representation that shows how total costs depend on various factors. In renting a car, the cost might depend on base rates and additional fees, such as mileage rates.
The cost equation for our car rental scenario is \(y = 45 + 0.25x\). Here, \(45\) is the fixed weekly rental fee, and \(0.25x\) represents the cost per mile traveled, where \(x\) is the number of miles.

• This type of equation helps calculate how expenses change based on usage.
• It's essential in budget discussions, as it clearly outlines cost components.

By identifying these elements, we can make informed decisions about our car rental expenses.

Problem Solving in Algebra

Problem solving in algebra involves translating real-world scenarios into mathematical equations or inequalities. The goal is to manipulate these equations to find solutions that meet specific conditions, like a budget.
Let's go through the steps:

• First, identify the problem and condition, such as staying within a $100 budget.
• Translate these into mathematical expressions, like the inequality: \(45 + 0.25x \leq 100\).
• Then, solve for the unknown variable, hoping to find how many miles you can drive.

This step-by-step approach allows for precise solutions that fit within the given parameters, providing clarity in decision-making.

Renting a Car Algebra Problem

A renting a car algebra problem often requires analyzing rental terms to determine costs based on usage. Here, it's crucial to understand both the fixed and variable costs within the cost equation.
For example, in the equation \(y = 45 + 0.25x\):

• \(\\(45\) is the fixed cost for renting the car weekly.
• \(\\)0.25x\) is the variable cost that depends on miles driven.

To solve how far you can drive on a budget, you substitute this equation into the linear inequality from your budget constraint, then solve for \(x\).
This method reveals how budget constraints impact possible travel distances, teaching essential budgeting skills.