Problem 36 - Applied Calculus

Question

You have a budget of 2000 dollars for the year to cover your books and social outings. Books cost (on average) 80 each and social outings cost (on average) 20 dollars each. Let \(b\) denote the number of books purchased per year and \(s\) denote the number of social outings in a year. (a) What is the equation of your budget constraint? (b) Graph the budget constraint. (It does not matter which variable you put on which axis.) (c) Find the vertical and horizontal intercepts, and give a financial interpretation for each.

Step-by-Step Solution

Answer

The budget constraint equation is \(80b + 20s = 2000\). The intercepts are (0 books, 100 outings) and (25 books, 0 outings). They represent the maximal expenditure on one category if none is spent on the other.

Step 1: Understand the budget constraint equation

Start with the total budget which is \(2000. You spend \)80 per book and \(20 per social outing. Considering the total cost of books is \)80b\( and the total cost of social outings is \)20s$, the budget constraint equation is formed as:\[ 80b + 20s = 2000 \]

Step 2: Simplify the budget constraint equation

To simplify, divide every term in the equation by 20:\[ 4b + s = 100 \]This is your simplified budget constraint equation.

Step 3: Identify the vertical intercept

Set the number of books, \(b\), to 0 to find the vertical intercept (when you don't purchase any books):\[ 4(0) + s = 100 \]\[ s = 100 \]This means you can afford 100 social outings if you buy no books.

Step 4: Identify the horizontal intercept

Set the number of social outings, \(s\), to 0 to find the horizontal intercept (when you don't go on any social outings):\[ 4b + 0 = 100 \]\[ b = 25 \]This means you can buy 25 books if you do not go on any social outings.

Step 5: Interpretation of intercepts

The vertical intercept represents spending the entire budget on social outings (100 outings), while the horizontal intercept represents spending the entire budget on books (25 books).

Step 6: Graph the budget constraint

To graph the budget constraint, place \(b\) on the x-axis and \(s\) on the y-axis. Plot the points (0, 100) for the vertical intercept and (25, 0) for the horizontal intercept. Connect these points with a straight line. This line represents all possible combinations of books and social outings you can afford within your $2000 budget.

Key Concepts

Graphing Equations

Graphing an equation is a visual representation of mathematical relationships between variables. In this case, our budget constraint equation describes how many books and social outings you can afford within a set budget.

• The equation in its simplified form is: \[ 4b + s = 100 \]
• This equation tells us that both the number of books \(b\) and social outings \(s\) can vary, but they must add up to fit within the budget.

Plotting this equation on a graph involves identifying two key intercepts, then connecting them with a straight line. The line shows all possible combinations of books and outings that use your entire budget.
Build understanding by practicing plotting equations as it helps in visualizing real-world scenarios. Graphing is a crucial aspect of interpreting linear equations in financial contexts.

Intercepts

Intercepts are points where the graph crosses the axes. They give crucial insights into financial scenarios and are easy to identify in budget constraints.

• Vertical Intercept: This occurs when the value of books \(b\) is zero, meaning all your budget goes to outings. The point is \[ (0, 100) \]
• Horizontal Intercept: This occurs when the value of outings \(s\) is zero, meaning all your budget is for books. The point is \[ (25, 0) \]

These intercepts are not just numbers but represent scenarios where you spend your entire budget on either books or outings. Understanding intercepts provides valuable insights into extremes of budget spending strategies. They also form the base points for creating the graph of the budget constraint.

Linear Equations

Linear equations like \[ 4b + s = 100 \]are simple yet powerful tools in representing relationships between two quantities. The beauty of a linear equation lies in its straightforward structure, making it ideal for modeling budget constraints.

• Slope: This aspect quantifies the rate at which one variable can change with respect to another while still maintaining the balance imposed by the budget.
• Graphing a linear equation results in a straight line expressing constant rates of trade-offs between variables.

Understanding linear equations allows one to predict and make informed decisions about allocating resources. Practically, they depict how much one might need to "sacrifice" one item to afford more of another within a constraint.

Financial Literacy

Financial literacy revolves around understanding how to manage money effectively. Applying concepts like budget constraints in real life can significantly enhance one's financial awareness.

• Using linear equations in budgeting empowers individuals to make informed decisions within their financial limits.
• Recognizing intercepts helps visualize extreme spending scenarios and better plan budgets.

With financial literacy skills, one can craft realistic financial plans and understand the impact of different choices. It's more than just numbers; it's about making strategic, informed, and confident decisions with your resources.