21 st Century Math Projects

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2 Project Title: Marketing Maverick Standard Focus: Algebra, Patterns, & Functions Time Range: 3-5 Days Supplies: TI-Graphing Technology Topics of Focus: - Quadratic Modeling - Solving Quadratic Equations - Finding the Vertex Benchmarks: Seeing Structure in Expressions Creating Equations Reasoning with Equations and Inequalities Interpreting Functions Interpreting Functions Interpreting Categorical and Quantitative Data Procedures: A-SSE A-CED A-REI F-IF F-IF S-ID 3b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 4. Solve quadratic equations in one variable. 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. 7a. Graph linear and quadratic functions and show intercepts, maxima, and minima. 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. A.) Students will complete Quadratic Modeling to begin practicing using quadratic models to solve real world problems. 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.) Students will complete Marketing Maverick in which they 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

3 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 to test 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 F(x) = x x+15 where x is the driving speed and F(x) is the fuel economy SUV F(x) = x x+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.

4 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 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.

5 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 are 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.

6 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 The cost per pound can be determined with the function C(x) = 0.004x x + 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 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 x 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: : : : : : : 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))

7 Pants Raw Materials The cost per pound can be determined with the function C(x) = x x where x = pounds of denim 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 Market research surveys have determined the most the average customer would be willing to pay for pants. The number of votes received can be estimated with the function V(x) = -x x 2250 where x is price in dollars. Determine the maximum price for the pants. Advertising Profit Based on the Re-Tweet data, decide when would be the best time to launch the Pinterest campaign? Time Re-Tweets 7: : : Our wholesaler is willing to purchase the shoes for $20 per unit. Your financial consultants expect the cost to be modeled by the function: C(x) = 0.01x 2 78x + 181,300 where x= number of units Write a profit function and determine the number of items you need to sell to maximize profit. 8: : : :00 94

8 Hoodie Raw Materials The cost per pound can be determined with the function C(x) = x x + 6 where x = pounds of cotton 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 Market research surveys have determined the most the average customer would be willing to pay for the hoodie. The number of votes received can be estimated with the function V(x) = -x x 1983 where x is price in dollars. Determine the maximum price for the hoodie. Advertising Profit Based on the Re-Tweet data, decide when would be the best time to launch the Pinterest campaign? Time Re-Tweets 7: : : Our wholesaler is willing to purchase the shoes for $32 per unit. Your financial consultants expect the cost to be modeled by the function: C(x) = 0.005x x 119,713 where x= number of units Write a profit function and determine the number of items you need to sell to maximize profit. 8: : : :00 190

9 Hats Raw Materials The cost per pound can be determined with the function C(x) = 0.001x 2.38x + 38 where x = pounds of wool 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 Market research surveys have determined the most the average customer would be willing to pay for the hat. The number of votes received can be estimated with the function V(x) = -x x where x is price in dollars. Determine the maximum price for the hat. Advertising Profit Based on the Re-Tweet data, decide when would be the best time to launch the Pinterest campaign? Time Re-Tweets 7: : : Our wholesaler is willing to purchase the shoes for $9 per unit. Your financial consultants expect the cost to be modeled by the function: C(x) = 0.08x 2 211x + 145,750 where x= number of units Write a profit function and determine the number of items you need to sell to maximize profit. 8: : : :00 671

10 marketing maverick Follow-Ups Use the results from your quadratic models to answer the following questions. 1) Which raw material were you able to get for the lowest price per pound? Rank the items from least expensive to most expensive. 2) Based on market research, which product do customers appear willing to spend the most money on? Which product do they appear least willing to spend? 3) When you meet with your Social Media Interns you will need to give them specific directions for Pinterest launch day. Write a schedule and tell them when to pin the products. 4) Which item do you expect to be most profitable? Rank the items from most profitable to least profitable. 5) You have an opportunity to highlight one item in a national television ad, which item will you choose and why?

11 Thank you for being my Math Friend! If you liked this 21 st Century Math Project You might like others. (Click the logo) Math it Up. Boomdiggy.

12 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 minimum 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 to test 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 F(x) = x x+15 where x is the driving speed and F(x) is the fuel economy SUV F(x) = x x+8 where x is the driving speed and F(x) is the fuel economy x=45.37 mph is the ideal speed which yields miles per gallon x=43.58 mph is the ideal speed which yields miles per gallon A) Is there a significant difference in the ideal speed of both automobiles? There is a 1.8 mph difference B) What is the difference in fuel economy between both automobiles? Which vehicle is better? There is a 7.05 miles per gallon difference. C) If you were running low on gas and you were traveling 50 mph, would you slow down or speed up? Support with evidence. You should slow down.

13 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. x = 190 units can be produced for $8.9 The cost per unit can be determined with the function C(x) = 0.003x 2 4.1x where x = number of units produced Determine the number of units that should be produced to achieve the lowest per unit cost. Where x = units can be produced for $19.17 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? $1691 would be the total cost of Widgets 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? $ would be total cost of the Thingamajigs 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. (240-2x)(x) = 240x 2 x 2 would be 7200 square feet with dimensions 60 x 120 A) If the rancher wants to be able to enclose 6,500 square feet of space, will she be able to? She will 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. No, she will be 5 feet short.

14 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? =(100+x)(975-8x) trees can be added which would have a maximum harvest of 98,457. A) Johnny Apple Farmer has a personal goal of harvesting 100,000 apples, will he meet that goal? Unfortunately, no he will not 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. He will have enough room. The Best Ticket price The Pro Soccer League has a trend of improved attendance on Friday nights, but are 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 y = x x y = = x x The maximum revenue of $206,613 is The maximum revenue of $360,928 is generated when tickets cost $ generated when tickets cost $ A) Is there a significant difference between the optimal price of tickets on Tuesdays and Fridays? There is slightly more than a $9 estimated difference between the optimal price on Tuesday and Friday. 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. Answers vary, but students should lean toward difference prices on different nights.

15 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 The cost per pound can be determined with the function C(x) = 0.004x x + 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 Market research surveys have determine the most the average customer would be willing to pay for the shoes. The number of votes received can be estimated with the function V(x) = -x x 4725 where x is price in dollars. Determine the maximum price for the shoes. The best unit price is 2.98 per pound if you order 72.5 pounds. $ would be the total cost. ~$75 is the maximum and it received 900 votes Advertising Profit Based on the Re-Tweet data, decide when would be the best time to launch the Pinterest campaign? y = x x The best time is approximately 9.7 to receive about repins. This would be at 9:42 Time Re-Tweets 7: : : : : : : Our wholesaler is willing to purchase the shoes for $40 per unit. Based on your financial consultants they expect the cost to be modeled by the function: C(x) = 0.05x 2 310x + 575,000 where x= number 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)) We need to sell 3500 units to have the highest earning potential which is $37,500 in profit.

16 Pants Raw Materials The cost per pound can be determined with the function C(x) = x x where x = pounds of denim 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 Market research surveys have determine the most the average customer would be willing to pay for the pants. The number of votes received can be estimated with the function V(x) = -x x 2250 where x is price in dollars. Determine the maximum price for the pants. The lowest price is $2.00 per pound if you purchase 401 pounds. The total cost would be $802 ~55 is the maximum with 775 votes Advertising Profit Based on the Re-Tweet data, decide when would be the best time to launch the Pinterest campaign? y = x x Best time is approximately 8.34 to receive about repins. This would be at 8:20. Time Re-Tweets 7: : : : : Our wholesaler is willing to purchase the shoes for $20 per unit. Based on your financial consultants they expect the cost to be modeled by the function: C(x) = 0.01x 2 78x + 181,300 where x= number of units Write a profit function and determine the number of items you need to sell to maximize profit. We need to sell 4900 units to have the highest earning potential which is $58,800 in profit. 9: :00 94

17 Hoodie Raw Materials The cost per pound can be determined with the function C(x) = x x + 6 where x = pounds of cotton 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 Market research surveys have determine the most the average customer would be willing to pay for the hoodie. The number of votes received can be estimated with the function V(x) = -x x 1983 where x is price in dollars. Determine the maximum price for the hoodie. ~$49 is the most popular with 418 votes The lowest unit price is 1.04 per pound if you purchase pounds. The total cost of the order would be $81.9 Advertising Profit Based on the Re-Tweet data, decide when would be the best time to launch the Pinterest campaign? y = x x Best time is approximately 7.46 to receive about repins. This would be at 7:28. Time Re-Tweets 7: : : : : Our wholesaler is willing to purchase the shoes for $32 per unit. Based on your financial consultants they expect the cost to be modeled by the function: C(x) = 0.005x x 119,713 where x= number of units Write a profit function and determine the number of items you need to sell to maximize profit. We need to sell 7850 units to have the highest earning potential which is $427,826 in profit. 9: :00 190

18 Hats Raw Materials The cost per pound can be determined with the function C(x) = 0.001x 2.38x + 45 where x = pounds of wool 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 Market research surveys have determine the most the average customer would be willing to pay for the hat. The number of votes received can be estimated with the function V(x) = -x x where x is price in dollars. Determine the maximum price for the hat. The best unit price for wool is $1.90 a pound if you purchase 190 pounds. The total cost would be $361. ~$20 is the most popular with 1210 votes Advertising Profit Based on the Re-Tweet data, decide when would be the best time to launch the Pinterest campaign? y = x x Best time is approximately 8.79 to receive about repins. This would be at 8:47. Time Re-Tweets 7: : : : : Our wholesaler is willing to purchase the shoes for $9 per unit. Based on your financial consultants they expect the cost to be modeled by the function: C(x) = 0.08x 2 211x + 145,750 where x= number of units Write a profit function and determine the number of items you need to sell to maximize profit. We need to sell 1375 units to have the highest earning potential which is $5,500 in profit. 9: :00 671

19 marketing maverick Follow-Ups Use the results from your quadratic models to answer the following questions. 1) Which raw material were you able to get for the lowest price per pound? Rank the items from least expensive to most expensive? 1.04 cotton, 1.90 wool, 2.00 denim, 2.98 leather 2) Based on market research, which product do customers appear willing to spend the most money on? Which product do they appear least willing to spend? $75 shoes and $20 hat. 3) When you meet with your Social Media Interns you will need to give them specific directions for Pinterest launch day. Write a schedule and tell them when to pin the products. Pin the items at the following times. 7:28 hoodie, 8:20 pants, 8:47 hat, 9:42 shoes 4) Which item do you expect to be most profitable? Rank the items from most profitable to least profitable. Hoodie $427,826, pants 58,800, shoes 37,500 hat 5,500 5) You have an opportunity to highlight one item in a national television ad, which item will you choose and why? It definitely has to be the hoodie given the possibility for its booming popularity. :0) Answers vary.