Marketing Maverick

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School

Colorado Early Colleges Parker *

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Course

1011

Subject

Mathematics

Date

May 14, 2024

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pdf

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Uploaded by kaitlynlandry270787

© Clark Creative Education Marketing Maverick Ideal Unit : Quadratic Functions Time Range : 3-5 Days Supplies : Graphing Technology, Pencil & Paper Topics of Focus: - Quadratic Modeling - Solving Quadratic Equations - Finding the Vertex Driving Question “How can quadratic functions aid in making business and marketing decisions?” Culminating Experience A marketing plan for a product launch. Common Core Alignment: A-SSE.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-REI.4 Solve quadratic equations in one variable. F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F-IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. S-ID.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models. Procedures: A.) In "Quadratic Modeling", students begin practicing using quadratic models to solve real world problems in five unique authentic contexts. Students will need to determine maximums or minimums. In some problems they will be given a function, in others they will have to write their own, and in the final they will have to have a regression model. B.) In "Marketing Maverick", students pretend to be a marketing director who has the responsibility to help launch a new clothing line. The 4 piece clothing line will have 4 problems each. For Raw Materials students will have to solve for a minimum value. In Market Research, they will solve for a maximum. For Advertising, students will need to use quadratic regression. In Profit, students will have to write their own Profit Function. Once they determine and interpret all of the correct maximums and minimums, they can complete “Marketing Maverick Follow-Ups”. * Aspects of the project can be completed independently. The entire project does not need to be completed to have a great learning experience, though it is suggested because it will best scaffold the skills and context.

© Clark Creative Education Quadratic Modeling It is no surprise that in business the goal is to make money. However, it could be surprising that producing more of a product may decrease profits. What if they don’t sell all of the product? What will they do with the extra? This leads to an important calculus term -- optimization . Optimizing results is the aim whether it is to achieve a maximum or a minimum. What is the ideal number of products to make to maximize profits? What is the ideal number of products to produce to minimize cost? While this is a business example, it is also relevant to personal financial matters and many other situations. In order to determine optimal results, quadratic modeling can be useful and we can explore the idea without using calculus… yet. In these situations, you will use quadratic models to help decision makers choose the best options. The Need for Speed??? In order to achieve the best gas mileage, there is science at work. In most automobiles driving too fast or too slow can have an effect on your fuel economy. A family has spent an afternoon testing the fuel economy of their vehicles at different speeds, and they have found quadratic models with data that they have collected. Help the family determine the ideal driving speeds of each of their vehicles that will save the most money. Sedan SUV F(x) = -0.00865x 2 +0.785x+15 where x is the driving speed and F(x) is the fuel economy F(x) = -0.00935x 2 +0.815x+8 where x is the driving speed and F(x) is the fuel economy A) Is there a significant difference in the ideal speed of both automobiles? B) What is the difference in fuel economy between both automobiles? Which vehicle is better? C) If you were running low on gas and you were traveling 50 mph, would you slow down or speed up? Support with evidence. Name ___________________________ Date ________________

© Clark Creative Education Shaving Production Cost Zagnabbit Electronics is reordering two critical pieces for their newest product. The goal is to minimize the cost for each unit, and they plan to purchase the quantity that has the lowest cost per unit. If they purchase too many, then they may incur extra labor, overtime or storage costs. Widgets Thingamajigs The cost per unit can be determined with the function C(x) = 0.001x 2 – .38x + 45 where x = number of units produced Determine the number of units that should be produced to achieve the lowest per unit cost. The cost per unit can be determined with the function C(x) = 0.003x 2 – 4.1x + 1420 where x = number of units produced Determine the number of units that should be produced to achieve the lowest per unit cost. A) For what quantity of Widgets is the per unit cost the lowest? If Zagnabbit placed an order for that quantity and price, how much would it be? B) For what quantity of Thingamajigs is the per unit cost the lowest? If Zagnabbit placed an order for that quantity and price, how much would it be? Build a Ranch A rancher has decided to build a rectangular fence in her backyard for her horses. The rancher purchased 240 feet of fence. The back of her home will be one side of the rectangle, and she wants to enclose the most land that she can. Help the rancher determine the maximum area she can enclose and the dimensions of the required fence. A) If the rancher wants to be able to enclose 6,500 square feet of space, will she be able to? B) The back of the rancher’s house is 55 feet long. Will it be enough to complete the entire rectangle? Support with evidence.

© Clark Creative Education Johnny Apple Farmer Johnny Apple Farmer has an orchard with 100 apple trees. Each tree produces 975 apples. Johnny has noticed that each time he plants a tree; the output per tree drops by 8 apples. How many trees should he add to the orchard in order to maximize the number of apples he can harvest? A) Johnny Apple Farmer has a personal goal of harvesting 100,000 apples. Will he meet that goal? B) Johnny Apple Farmer estimates he can plant another 15 trees on his land. Will that be a problem? Support with evidence. The Best Ticket price The Pro Soccer League has a trend of improved attendance on Friday nights, but is struggling to draw crowds on the weekdays. The PSL has considered a plan to offer discounted tickets on weekday games and have recently tested their idea across the country. They have collected data on Tuesday and Friday games and created mathematical models to help them figure out the best prices of their tickets to maximize revenue. Help the PSL determine if there is a significant difference between the optimal ticket price of Tuesday and Friday games and help them make a decision for their league. Tuesday Friday Ticket Price (x) Total Revenue Generated (y1) Total Revenue Generated (y2) $10 $178,000 $178,000 $15 $198,000 $268,000 $20 $205,000 $331,000 $25 $180,000 $356,000 $30 $135,000 $358,000 $35 $54,000 $320,000 Function A) Is there a significant difference between the optimal price of tickets on Tuesdays and Fridays? B) Would you suggest that the PSL have ticket prices that vary on different nights or should they have a consistent single price? Support with evidence.

© Clark Creative Education Marketing Maverick Swagg Suit, Inc. is ready to launch a new clothing line, and since you are the Director of Marketing, your plate just got full. Focus group testing, social media campaigns and maximizing profits are now on your to-do list. Lucky for you, you have a solid grasp of quadratic modeling and can use that evidence to help you make decisions. (Isn’t that what you said in your job interview?) Your CEO has scheduled a meeting with you to discuss the launch as soon as possible, and she is expecting your recommendations. Shoes Raw Materials Market Research The cost per pound can be determined with the function: C(x) = 0.004x 2 – 0.58x + 24 where x = pounds of leather Determine the number of pounds that should be purchased to achieve the lowest per pound cost and the total cost to place that order. Market research surveys have determined the most the average customer would be willing to pay for shoes. The number of votes received can be estimated with the function: V(x) = -x 2 + 150x – 4725 where x is price in dollars Determine the maximum price for the shoes. Advertising Profit Based on the Re-Tweet data, decide when would be the best time to launch the Pinterest campaign. Time Re-Tweets 7:00 115 7:30 220 8:00 301 8:30 351 9:00 382 9:30 410 10:00 408 Our wholesaler is willing to purchase the shoes for $40 per unit. Your financial consultants expect the cost to be modeled by the function: C(x) = 0.05x 2 – 310x + 575,000 where x = # of units Write a profit function and determine the number of items you need to sell to maximize profit. (Hint P(x) = R(x) – C(x)) Name ___________________________ Date ________________

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