Problem 21 - Intermediate Algebra

Question

Moving. To help move his furniture, a man rents a truck for S41.50 per day plus 35 \(\notin\) per mile. If he has budgeted \(\$ 150\) for transportation expenses, how many miles will he be able to drive the truck if the move takes 1 day?

Step-by-Step Solution

Answer

He can drive approximately 310 miles.

Step 1: Determine Fixed Costs

First, identify the fixed cost given that the man will rent the truck for one day. The daily rental fee is $41.50.

Step 2: Calculate Available Budget for Miles

Subtract the fixed cost from the total budget to determine the available budget for driving miles. This calculation is: \(150 - 41.50 = 108.50\). So, he has \(\$108.50\) remaining for mileage.

Step 3: Determine Cost Per Mile

The cost per mile of driving the truck is 35 cents, or \(0.35\) dollars per mile.

Step 4: Calculate Maximum Miles

To find out how many miles he can drive, divide the available budget for miles by the cost per mile:\[ \text{Maximum miles} = \frac{\text{Available budget for miles}}{\text{Cost per mile}} = \frac{108.50}{0.35} \approx 310 \, \text{miles} \]

Key Concepts

Understanding Fixed and Variable Costs

Fixed and variable costs are key concepts in budgeting and economic planning. In the context of renting a truck, fixed costs refer to the expense that remains the same no matter how much or how little the truck is used. Here, the fixed cost is the daily rental fee of $41.50.

On the other hand, variable costs change depending on usage. For instance, in this exercise, the cost per mile, priced at 35 cents or $0.35, is variable. These costs increase as the truck is driven more miles.

This differentiation helps in calculating expenses more accurately. When planning a budget, it's crucial to distinguish between these fixed expenses that will occur regardless of usage, and variable costs that depend on it. This understanding aids in precise financial forecasting and efficient resource allocation.

Essentials of Budget Management

Budget management is about planning and controlling income and expenses. It allows individuals and organizations to maintain financial health and meet their financial goals.

In our exercise, the overall budget for transportation is \(150. The key steps involve:

• Identifying all costs: Here, include both fixed costs (the daily truck rental) and variable costs (per mile charge).
• Allocating funds efficiently: By subtracting the fixed costs from the total budget, we determine what's left for variable expenses. In this case, \(150 - 41.50 \) leaves \)108.50 for mileage.
• Making informed decisions: With a clear budget outline, you can calculate the maximum number of miles the man can afford, preventing overspending.

Mastering these steps ensures that expenses do not exceed the available budget, promoting financial stability.

Fundamentals of Mathematical Problem Solving

Mathematical problem solving is a critical skill, not just for academics but for real life. It involves logical thinking and quantitative reasoning to find solutions.

The process begins with understanding the problem. For our exercise, it means knowing the budget constraint and the costs involved.

• Identify the unknown: Here, it's the maximum miles that can be driven.
• Establish a relationship: Using the cost per mile, create an equation representing the budget constraints (\(150 - 41.50 = 108.50\) for miles).
• Solve systematically: Divide the available funds for mileage by the cost per mile (\(\frac{108.50}{0.35}\) to determine the answer).
• Verify and interpret: Check your math and ensure the solution makes sense within the given budget.

By practicing these methods, students enhance their ability to approach complex problems with confidence and accuracy.