Question

Petra has \(\$ 480\) to spend on DVDs and books. A book costs \(\$ 24\) and a DVD costs \(\$ 15\). [LO 7.2] a. Write an equation for the budget constraint. Let \(x=\) books. Let \(y=\) DVDs. b. Use your equation to determine how many books Petra can buy if she buys 8 DVDs.

Short answer

Petra can buy 15 books if she buys 8 DVDs.

Step-by-step solution

Define the Budget Constraint Equation

To set up the equation, we consider the total amount Petra has, which is \(\\(480\). The cost of each book is \(\\)24\) and each DVD is \(\$15\). We can write the budget equation as: \[ 24x + 15y = 480 \]where \(x\) represents the number of books and \(y\) represents the number of DVDs.

Substitute the Number of DVDs

We need to find out how many books Petra can buy if she purchases 8 DVDs. Substitute \(y = 8\) into the budget equation:\[ 24x + 15(8) = 480 \]

Simplify the Equation

Calculate the total cost for 8 DVDs:\[ 15 \times 8 = 120 \]Substitute this value back into the equation:\[ 24x + 120 = 480 \]

Solve for x

Subtract 120 from both sides to isolate the terms with \(x\):\[ 24x = 480 - 120 \]\[ 24x = 360 \]Divide both sides by 24 to find \(x\):\[ x = \frac{360}{24} = 15 \]

Key concepts

Linear Equations

Linear equations are pivotal in representing relationships between different quantities. In Petra's scenario, her purchasing choices for books and DVDs can be modeled using a linear equation. The general format for a linear equation involving two variables, like in our case, is \( ax + by = c \), where \( a \) and \( b \) are coefficients, \( x \) and \( y \) are variables, and \( c \) is a constant. Here, each book costs \(\\(24\) and each DVD costs \(\\)15\), leading to the equation \( 24x + 15y = 480 \). This equation is called a budget constraint, showing all possible combinations of books and DVDs Petra can purchase without exceeding her budget.

Solving the equation involves basic algebraic operations like substitution and elimination to determine the possible values of \(x\) and \(y\). The goal is to find the feasible combinations of purchasing books and DVDs respecting the budget limitation. In our case, when the number of DVDs (\(y\)) is fixed, we can easily determine the number of books (\(x\)) Petra can purchase.

Consumer Choice

Consumer choice explores how individuals like Petra make decisions to allocate their limited resources, like income, among various alternatives to maximize utility or satisfaction. In our exercise, Petra considers two types of goods - books and DVDs. Her choice depends on her preferences and the prices of these items.

The equation \( 24x + 15y = 480 \) restricts Petra’s choices, as it represents the combinations of books and DVDs that fit within her \(\$480\) budget. The number of goods she can buy depends on the price of each and her overall budget, illustrating the essence of consumer choice theory.

If we learn how many DVDs Petra plans to purchase (like 8 DVDs here), the equation helps us understand how many books are within her remaining budget. This reflects the constant trade-offs that consumers face.

• It helps in assessing the impact of changing prices of goods on the consumer's ability to purchase them.
• Consumers aim to maximize their satisfaction with the budget they have, which involves estimating the utility from different combinations of goods.

Opportunity Cost

Opportunity cost is a fundamental concept in economics that refers to the value of the next best alternative forgone. It's crucial in decision-making processes, including consumer choice like Petra's decision on how to spend her \(\\(480\).

When Petra decides to buy 8 DVDs, each decision involves an opportunity cost because resources used for one item can no longer be used for another. For instance, each time Petra buys a DVD for \(\\)15\), she sacrifices the opportunity to spend that money on books. If books are priced at \(\$24\), purchasing a DVD represents an implicit trade-off of not buying part of a book.

This concept is crucial as it underlines the cost assessment consumers undertake, where the opportunity cost of choosing more DVDs means fewer books, impacting Petra's way of maximizing her satisfaction from her spending.

It highlights decisions within the context of limited resources, emphasizing how choice inevitably involves sacrifice, balanced by personal preferences and utility maximization strategies.