Question

MULTI-STEP PROBLEM A customer of your flower shop wants to send flowers to 23 people. Each person will receive an \(\$ 11.99\) "sunshine basket" or a \(\$ 16.99\) "meadow bouquet." a. Let \(s\) represent the number of people who will receive a sunshine basket. Which function can you use to find \(C\), the total cost of sending flowers to all 23 people, depending on how many of each arrangement is sent? (A) \(C=16.99(23-s)+11.99 s\) (B) \(C=11.99 s+16.99(23)\) b. If 8 people receive a sunshine basket, what is the total cost of the flowers? c. If 13 people receive a meadow bouquet, what is the total cost of the flowers? d. CRITICAL THINKING If your customer can spend only \(\$ 300\), what is the greatest number of people that can receive a meadow bouquet?

Short answer

a. The correct cost function is (A) \(C=16.99(23-s)+11.99s\). b. If 8 people receive sunshine baskets, the total cost will be \$381.83. c. If 13 people receive meadow bouquets, the total cost will be \$356.77. d. The maximum number of people who can receive a meadow bouquet for a budget of \$300 is 17.

Step-by-step solution

Identify the correct function for total cost

We're told that there are two types of bouquets: the sunshine basket, costing \(11.99\) and the meadow bouquet, costing \(16.99\). The total number of people receiving flowers is 23. If \(s\) people receive the sunshine basket, then \(23 - s\) people receive the meadow bouquet. So, the correct function for total cost based on number of sunshine baskets, \(s\), is \(C=16.99(23-s)+11.99s\). Hence, answer choice (A) is correct.

Calculate the total cost if 8 people receive a sunshine basket

Substitute \(s=8\) into the function to calculate the total cost: \(C=16.99(23-8)+11.99*8\), which simplifies to \(C = 381.83\).

Calculate the total cost if 13 people receive a meadow bouquet

If 13 people receive the meadow bouquet, that means the rest (23-13=10) people receive the sunshine basket. Substitute \(s=10\) into function to calculate the total cost: \(C=16.99(23-10)+11.99*10\), which simplifies to : \(C = 356.77\).

Calculate the maximum number of people that can receive a meadow bouquet with a budget of $300

If the customer can only spend \(300\), we need to find the greatest number of people that can receive the more expensive meadow bouquet. For this, calculate the cost of the meadow bouquet for each person from 1 to 23 until the total cost exceeds \(300\). Once you calculate that sending the meadow bouquet to 16 people results in a total cost of \(271.84\) and sending the meadow bouquet to 17 people results in a total cost of \(288.83\), you will realize that the customer can only send the meadow bouquet to 17 people.

Key concepts

Systems of Equations

Understanding how systems of equations work is fundamental in algebra. Essentially, a system of equations involves two or more equations set equals to each other that share some of the same variables.

In the context of our exercise, we can imagine having different equations for each type of bouquet cost, which when combined, gives us a system that can be managed through algebraic manipulation. For example, let's say we have two different types of arrangements: the sunshine basket and the meadow bouquet. The system of equations would consist of one equation that calculates the cost of sunshine baskets and another that calculates the cost of meadow bouquets. By adding them together, we can solve for the total cost for all bouquets sent to 23 people.

Working with systems of equations is not just about solving for a variable—it's about understanding the relationships between the variables and how changing one variable affects the other(s). It requires a balance, much like a scale, to ensure that the adjustments made on one side of the equation does not overpower the other, keeping the system consistent.

Linear Equations

Linear equations form the backbone of algebra and are used to describe a range of real-world situations with a straight-line relationship between variables. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.

In our example, the total cost of the flowers can be considered a linear equation, where the cost is a function of the number of each type of bouquet sent. These equations are incredibly versatile and are a starting point for solving more complex algebraic equations. It's important to grasp that linear equations allow us to predict one variable, given another—for instance, calculating the number of bouquets of each type from the total cost. Knowing how to formulate and solve these equations is crucial because it's not just about finding an answer, but understanding the process and relationship between the variables involved.

Cost Analysis

Cost analysis in algebra is a practical application of the discipline in a business context. It allows for the assessment of different financial options and decisions based on algebraic models. Perfect for decision-making, cost analysis involves setting up algebraic equations to represent different financial scenarios.

In our flower shop problem, cost analysis is used to optimize the number of bouquets a customer can send within a specified budget. It requires us to understand the linear relationship between the cost of each bouquet and the total number of recipients. By calculating these costs, we can also plan for spending and savings. It's this kind of analysis that takes algebra out of the classroom and into the real world, emphasizing that understanding algebra can lead to better financial decisions.

Critical Thinking in Algebra

Algebra isn't just about solving for x; it's a way of thinking critically about problems and finding logical solutions. Critical thinking in algebra involves breaking down complex problems into simpler parts, analyzing the relationship between variables, and choosing the appropriate methods to solve a question.

For the critical thinking part of our exercise, the customer's budget limitation requires strategic decision-making. By examining the constraints (the budget) and the available options (type and cost of bouquets), one needs to algebraically determine the maximum number of people who can receive the meadow bouquet. Students must therefore engage with the problem, exploring various possibilities and extracting meaningful answers that are not just mathematically correct but also make sense in a real-life context.