Problem 65 - Algebra A Combined Function

Question

Dennis and Nancy Wood are celebrating their 30 th wedding anniversary by having a reception at Tiffany Oaks reception hall. They have budgeted \(\$ 3000\) for their reception. If the reception hall charges a \(\$ 50.00\) cleanup fee plus \(\$ 34\) per person, find the greatest number of people that they may invite and still stay within their budget.

Step-by-Step Solution

Answer

Dennis and Nancy can invite a maximum of 86 people.

Step 1: Define the Equation

Let the number of people Dennis and Nancy invite be represented by \( n \). The total cost charged by the reception hall includes a cleanup fee of \( 50 \) dollars and an additional \( 34 \) dollars per person. Therefore, the cost equation is \( 34n + 50 \).

Step 2: Establish the Budget Constraint

Dennis and Nancy have budgeted \( 3000 \) dollars for their reception. Therefore, the total cost \( 34n + 50 \) must be less than or equal to \( 3000 \). The inequality representing this constraint is: \[ 34n + 50 \leq 3000 \]

Step 3: Solve for the Maximum Number of People

To solve for \( n \), subtract \( 50 \) from both sides of the inequality: \[ 34n \leq 2950 \]Next, divide both sides by \( 34 \): \[ n \leq \frac{2950}{34} \] Calculating the division gives \( n \leq 86.7647 \). Since \( n \) must be an integer, round down to 86.

Key Concepts

Budget Constraint

The concept of a budget constraint is crucial for any financial planning, whether personal or organizational. It represents the limits imposed by available funds. In other words, it defines what you can afford to spend based on your budget. In this exercise, the budget constraint is clearly set at $3000, representing the maximum amount Dennis and Nancy Wood are willing to spend on their anniversary reception.

Setting a budget constraint works like a guideline, ensuring you don't overspend. Think of it as a financial "frame" holding all your spending choices. For Dennis and Nancy, the constraint ensures that every decision they make about their reception fits within their financial plan.

If they want to invite more people, they must consider whether the cost per person still keeps them under their budget. By viewing their expenses in this way, they're making informed choices and avoiding potential overspending.

Linear Equation

Linear equations are fundamental in algebra and serve as a straightforward mathematical expression to relate different quantities. In most everyday scenarios, like budgeting, a linear equation can help us calculate relationships between variables.

In the given exercise, the cost equation is linear, expressed as \(34n + 50\). This equation adds together the constant cleanup fee (\(50dollars) with a variable cost per guest (\)34). Here, \(n\) refers to the number of people invited. This equation allows Dennis and Nancy to understand the total reception cost as they adjust the number of guests, \(n\).

Understanding linear equations makes it easy to solve for unknowns when other variables are given. In this case, Dennis and Nancy adjust *n* until their cost equals their budget limit, finding out their maximum allowable guests.

Integer Solutions

Sometimes in real-world problems, solutions need to be practical and applicable. This often means dealing with whole number solutions, as fractions or decimals wouldn’t fit the scenario. Integer solutions, like in our exercise, often arise from these situations.

In tackling the given inequality \(34n + 50 \leq 3000\), solving for \(n\) resulted in \(n \leq 86.7647\). However, you cannot invite a fraction of a person to a party!

Thus, the practical application of integer solutions comes into play, where only whole numbers make sense. Dennis and Nancy must invite 86 people, rounding down to the nearest whole number because it fits their budget perfectly without exceeding it. Integer solutions therefore provide viable, actionable answers aligned with real life.