Problem 19 - Algebra 2

Question

RECORDING Jane's band wants to spend no more than \(\$ 575\) recording their first \(C D\) . The studio charges at least \(\$ 35\) an hour to record. Graph a system of inequalities to represent this situation.

Step-by-Step Solution

Answer

Jane's band can record for a maximum of 16 hours without exceeding \( \$575 \).

Step 1: Define the Variables

Let \( h \) be the number of hours Jane's band spends in the studio. Each hour costs at least \( \\(35 \), and the total expenditure should be less than or equal to \( \\)575 \).

Step 2: Establish the Inequality for Cost

The cost for \( h \) hours is given by \( 35h \). Since the band cannot spend more than \( \$575 \), we have the inequality \( 35h \leq 575 \).

Step 3: Solve the Inequality

To solve for \( h \), divide both sides of the inequality by 35 to get \( h \leq \frac{575}{35} = 16.43 \). Since hours must be whole numbers, the band can spend at most 16 hours recording.

Step 4: Establish the Inequality for Studio Charge

Since the studio charges at least \( \$35 \) per hour, we have the inequality \( 35 \leq \) hourly rate. However, as hourly rate directly impacts the total cost, this does not affect the graph since the hourly rate is already included in the calculation of the cost inequality.

Step 5: Graph the Inequalities

On a graph, let the x-axis represent hours (\( h \)) and y-axis represent cost. Plot the line \( y = 35x \). The region below this line represents potential values for \( 35h \leq 575 \); shade this region. Since \( h \) must be an integer, the potential solutions are points on the x-axis where \( h \leq 16 \). The shaded area forms a triangle under the line extending up to \( h = 16 \).

Key Concepts

Graphing Inequalities

Graphing inequalities involves visually representing the relationship between variables within a mathematical inequality. In Jane’s scenario with the recording studio, start by plotting potential outcomes on a graph with hours (\( h \)) on the x-axis and total cost on the y-axis. To graph the inequality \( 35h \leq 575 \), first determine the line \( y = 35h \).

This line defines the maximum potential cost at each given hour. Next, you shade the area below this line because this region represents all combinations of hours and associated costs that meet the constraint. Only points where \( h \) is a whole number (0, 1, 2,..., 16) are valid solutions in this context.

The shaded area helps clearly see a range of values that satisfy the inequality. This visual depiction not only aids comprehension but also provides a quick reference for decision making.

Cost Constraints

Cost constraints define the financial limits within which a decision must be made. For Jane's band, the total budget is \(\(575\). To express this constraint mathematically, we examine the costs associated with hiring the recording studio. The studio charges a minimum of \(\)35\) per hour.

Given this structure, the inequality \(35h \leq 575\) captures the essence of staying within budget. A key feature of cost constraints is that they guide financial planning, ensuring that expenses do not exceed the available funds.

• They help in prioritizing expenses.
• They enforce limits on spending time in the studio.

These constraints are like a safety net, preventing overspending and ensuring better financial management.

Linear Inequalities

Linear inequalities are expressions where the relation between variables is not equals, making them different from simple linear equations. In Jane's case, \(35h \leq 575\) describes a linear inequality. It represents a condition where the total studio cost must not cross a certain limit.

Solving linear inequalities follows a similar approach to solving equations, but with key differences when it comes to directionality. Here, dividing both sides by 35 gives \(h \leq 16.43\). Since hours must be whole numbers, it implies \(h \leq 16\).

Working with linear inequalities:

• Understand the signs (\(<, >, \leq, \geq\)) which determine boundary conditions.
• Convert the inequalities to represent feasible solutions on a graph.

These steps enable students to tackle a broad range of real-world problems by assessing feasible solutions.