Systems of equations are used in many cost models in the real world.
Systems of equations are used in many cost models in the real world. Here is an example scenario for you to consider.
• Suppose you are running a carnival. You are selling hamburgers and sodas. A hamburger is $1.75 and a soda is .75.
• You expect to make a total of $117.50 for the day
• You also plan to sell 120 hamburgers and sodas
• How many sodas and hamburgers will you sell each?
• Suppose you decide to change the price ratio between hamburgers and soda’s so that they produce a more equal consumption of hamburgers and sodas.
o What would you change the price to for each and why?
o What would be the new amount of hamburgers and soda each at your price points?
• Consider your peer’s example.
o How does it compare to the scenario provided?
o How does their example contribute to your understanding of systems of equations?
• Is their example something you might see in your profession as well?
o How would it be similar?
o How would it be different?
• Did the changes to the equations make sense to you? Why or why not?
Please be sure to validate your opinions and ideas with citations and references in APA format.
Estimated time to complete: 1 hour
Unit 3
Suppose you work for a primary care clinic in peak influenza season. Each pre-dosed vaccine costs $15 and their nurse visit costs $20. At the end of your clinic day, you made $2,000. You saw 55patients.
The equation would be
15v+ 20n=2000
The quantity equation would be
V+n=55
Use the substitution method
15(55- n)+20n=2000
Simplify and solve for n
825-15n+20n=2000
-15n+20n=5n
825+5n=2000
Subtract 825 from both sides
5n=1175
Divide by 5
N=235
Substitute N back into the quantity equation to find V
V+235=55
V=235-55
Subtract
V= 180
Sample Solution
Carnival Scenario: Analyzing Cost Models with Systems of Equations
1. Carnival Profits and Equations:
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Given: Hamburger price = $1.75, Soda price = $0.75, Target revenue = $117.50, Total items sold = 120 (hamburgers+sodas).
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Problem: Find the number of hamburgers and sodas sold.
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Solution: This can be modeled using a system of two equations:
- Equation 1: 1.75H + 0.75S = 117.50 (Revenue equation)
- Equation 2: H + S = 120 (Total items equation)
Full Answer Section
Solving this system using substitution or elimination methods will give you: * H = 75 hamburgers * S = 45 sodas- Rebalancing Consumption with Price Adjustments:
- Goal: Change price ratio to encourage more balanced consumption of hamburgers and sodas.
- Strategy: Increase soda price relative to hamburgers, making them closer in cost.
- New Prices: (Possible options)
- Option 1: Hamburger = $1.75, Soda = $1.00
- Option 2: Hamburger = $1.50, Soda = $0.90
- New Sales Calculation: With the new prices, you would need to recalculate the number of hamburgers and sodas sold using the same system of equations with updated price values.
- Comparing Your Peer's Example:
- Similarities: Both scenarios likely involve using systems of equations with price and quantity variables to model a real-world cost scenario.
- Differences: The specifics of the scenario (industry, products, prices, goals) will differ. Your carnival example focuses on balancing consumption, while your peer's example might have a different objective, such as maximizing profit or meeting specific demand conditions.
- Relevance to Your Profession: Systems of equations are versatile tools applicable in various professions, including healthcare, finance, business, and engineering. They allow you to model and analyze complex relationships between variables and make informed decisions based on the results.
- Changes to the Equations:
- Citations and References:
- Chang, A. T., & Gerald, K. F. (2015). Applied mathematical modeling for business and management. John Wiley & Sons.
- Nagle, T. T., & Martin, S. E. (2018). Decisionmaking with marketing models. Pearson.
- Shahian, F., & Smith, G. (2021). Systems of equations: Algebra II: A complete introduction. Independently published.