You are starting a new business in which you have decided to sell two products instead of just one. Determine a business you could start and choose two products that you could sell. How much of your own money are you willing to invest in this business in order to get started? How much will each item cost you to make? How much will you charge for each item? a. Explain this business (how much of your own money you’re willing to spend on the business, what items you’re going to sell, costs for each item, sale price for each item, etc.). b. Consider the total amount you’re willing to spend on the business and how much it will cost you to make your items. Write an inequality that represents the fact that while making each item, you can’t exceed this limit. Be sure to include the cost per item in this inequality. c. Graph your inequality. Be sure to label your graph and shade the appropriate side of the line. d. Choose a point that falls in the shaded region. Explain what the x-coordinate and y-coordinate represent and the significance in terms of cost of this point falling in the shaded region. e. Choose a point that falls directly on the line. Explain what the x-coordinate and y-coordinate represent and the significance in terms of cost of this point falling directly on the line. f. Choose a point that does not fall in the shaded region. Explain what the xcoordinate and y-coordinate represent and the significance in terms of cost of this point falling outside of the shaded region. © 2014 Connections Education LLC. All rights reserved. Task 2 Consider the total amount you’re willing to spend to start your business. After selling your items, you want your total amount earned to be at least three times the amount you originally spent. a. How much money are you hoping to earn from selling your products? b. Determine the price you want to sell each item for. Note:You may need to adjust the original prices that you came up with in Task 1. c. Write an inequality that represents the fact that you want your total earned to be at least three times the amount that you originally spent. Be sure to include the price for each item in this inequality. d. Graph your inequality. Be sure to label your graph and shade the appropriate side of the line. In the context of the problem, does the shaded area make sense? If not, make sure to adjust the original values (the investment and the sale price for both products) so that your shaded area is reasonable. e. Choose a point that falls in the shaded region for both this inequality and the inequality you found in Task 1. Explain what the x-coordinate and ycoordinate represent and the significance in terms of both cost and money earned of this point falling in the shaded regions. f. What is the ideal number of items you should produce and sell? Why? Task 3 Pick one of your items. You have been contacted by a customer in Alaska who wants to purchase several of these items and wants you to ship the items to her. You have already established the cost per item and you will only charge the customer $5 to ship these items to Alaska. Suppose another company sells the same item but charges half of your price. However, if the customer buys from this company, she will be charged $20 in shipping costs. a. Write two equations to represent the customer’s total cost based on how many items she buys from each of the two sellers—you and the other company. b. If the customer in Alaska wants to buy 5 items, from whom should she buy? Explain your answer. c. If the customer in Alaska wants to buy 50 items, from whom should she buy? Explain your answer. d. Solve the system of equations from part A. What method did you choose to solve the system? Why? e. Explain what your solution for part D means in terms of the situation.
a. One possible business idea could be selling handmade jewelry and accessories. I am willing to invest $5,000 of my own money to get started. The cost to make each jewelry item is $10, and the cost to make each accessory item is $5. I will sell each jewelry item for $30 and each accessory item for $15.
b. The total amount I am willing to spend on the business is $5,000. The cost per jewelry item is $10, and the cost per accessory item is $5. We can write the inequality as: 10x + 5y ≤ 5000, where x represents the number of jewelry items and y represents the number of accessory items.
c. The graph of the inequality will have x and y on the axes. The x-axis will represent the number of jewelry items, and the y-axis will represent the number of accessory items. The shaded region will be below and on the line 10x + 5y = 5000.
d. A point that falls in the shaded region could be (250, 700). The x-coordinate represents the number of jewelry items (250), and the y-coordinate represents the number of accessory items (700). The significance is that this point falls within the affordable range of the total investment cost.
e. A point that falls directly on the line could be (500, 0). The x-coordinate represents the number of jewelry items (500), and the y-coordinate represents the number of accessory items (0). The significance is that at this point, the total investment cost is exactly the amount I am willing to spend.
f. A point that does not fall in the shaded region could be (100, 600). The x-coordinate represents the number of jewelry items (100), and the y-coordinate represents the number of accessory items (600). The significance is that this point exceeds the affordable range of the total investment cost.
Task 2:
a. I am hoping to earn at least $15,000 from selling my products.
b. To determine the price I want to sell each item for, I need to adjust the original prices. I will sell each jewelry item for $35 instead of $30, and each accessory item for $20 instead of $15.
c. The inequality representing the desire to earn at least three times the original investment is: 35x + 20y ≥ 3(5000) = 15000, where x represents the number of jewelry items and y represents the number of accessory items.
d. The graph of the inequality will have x and y on the axes. The x-axis will represent the number of jewelry items, and the y-axis will represent the number of accessory items. The shaded region will be above and on the line 35x + 20y = 15000. The shaded area makes sense in this context as it represents the region where the total earnings are at least three times the original investment.
e. A point that falls in the shaded region for both inequalities could be (300, 300). The x-coordinate represents the number of jewelry items (300), and the y-coordinate represents the number of accessory items (300). The significance is that at this point, the total earning exceeds three times the original investment.
f. The ideal number of items to produce and sell depends on various factors such as market demand, production capacity, and competition. However, based on the given information, producing and selling around 300 jewelry items and 300 accessory items would be ideal as it falls within the shaded region and helps achieve the goal of earning at least three times the original investment.
Task 3:
a. The total cost for buying from me would be 5x + 5, where x represents the number of items bought. The total cost for buying from the other company would be 0.5x + 20, where x represents the number of items bought.
b. If the customer in Alaska wants to buy 5 items, she should buy from the other company. The cost of buying from the other company would be 0.5(5) + 20 = $22.5, while the cost of buying from me would be 5(5) + 5 = $30.
c. If the customer in Alaska wants to buy 50 items, she should buy from me. The cost of buying from me would be 5(50) + 5 = $255, while the cost of buying from the other company would be 0.5(50) + 20 = $45.
d. To solve the system of equations, we can set the two costs equal to each other: 5x + 5 = 0.5x + 20. By solving this equation, we find that x = 3.75. We can round up to 4, which means the customer should buy 4 items from the other company.
I chose to solve the system of equations algebraically by setting the two costs equal to each other. This method allows me to find the specific quantity of items at which the costs are equal.
e. The solution for part d (x = 4) means that if the customer buys 4 items, the cost will be the same regardless of whether she buys from me or the other company. This point represents the break-even point for the customer, where the total cost is the same for both options.
f. The ideal number of items the customer should buy depends on factors like the customer's budget, preference, and urgency. However, based on the comparison of costs, if the customer wants to minimize the total cost, she should buy 4 items from the other company, as it will be the same cost as buying from me.
what about a tech buisness
a. In the tech business, I am willing to invest $50,000 of my own money to get started. I have decided to sell smartphones and smartwatches. The cost to make each smartphone is $200, and the cost to make each smartwatch is $100. I will sell each smartphone for $500 and each smartwatch for $250.
b. The total amount I am willing to spend on the business is $50,000. The cost per smartphone is $200, and the cost per smartwatch is $100. We can write the inequality as: 200x + 100y ≤ 50000, where x represents the number of smartphones and y represents the number of smartwatches.
c. The graph of the inequality will have x and y on the axes. The x-axis will represent the number of smartphones, and the y-axis will represent the number of smartwatches. The shaded region will be below and on the line 200x + 100y = 50000.
d. A point that falls in the shaded region could be (100, 250). The x-coordinate represents the number of smartphones (100), and the y-coordinate represents the number of smartwatches (250). The significance is that this point falls within the affordable range of the total investment cost.
e. A point that falls directly on the line could be (250, 0). The x-coordinate represents the number of smartphones (250), and the y-coordinate represents the number of smartwatches (0). The significance is that at this point, the total investment cost is exactly the amount I am willing to spend.
f. A point that does not fall in the shaded region could be (80, 400). The x-coordinate represents the number of smartphones (80), and the y-coordinate represents the number of smartwatches (400). The significance is that this point exceeds the affordable range of the total investment cost.
Task 2:
a. I am hoping to earn at least $150,000 from selling my products.
b. To determine the price I want to sell each item for, I need to adjust the original prices. I will sell each smartphone for $600 instead of $500, and each smartwatch for $300 instead of $250.
c. The inequality representing the desire to earn at least three times the original investment is: 600x + 300y ≥ 3(50000) = 150000, where x represents the number of smartphones and y represents the number of smartwatches.
d. The graph of the inequality will have x and y on the axes. The x-axis will represent the number of smartphones, and the y-axis will represent the number of smartwatches. The shaded region will be above and on the line 600x + 300y = 150000. The shaded area makes sense in this context as it represents the region where the total earnings are at least three times the original investment.
e. A point that falls in the shaded region for both inequalities could be (150, 300). The x-coordinate represents the number of smartphones (150), and the y-coordinate represents the number of smartwatches (300). The significance is that at this point, the total earning exceeds three times the original investment.
f. The ideal number of items to produce and sell depends on various factors such as market demand, production capacity, and competition. However, based on the given information, producing and selling around 150 smartphones and 300 smartwatches would be ideal as it falls within the shaded region and helps achieve the goal of earning at least three times the original investment.
Task 3:
a. Let's assume the customer wants to buy x number of items. The total cost for buying from me would be 200x + 5, where x represents the number of items bought. The total cost for buying from the other company would be 100x + 20, where x represents the number of items bought.
b. If the customer wants to buy 5 items, she should buy from the other company. The cost of buying from the other company would be 100(5) + 20 = $520, while the cost of buying from me would be 200(5) + 5 = $1005.
c. If the customer wants to buy 50 items, she should buy from me. The cost of buying from me would be 200(50) + 5 = $10,005, while the cost of buying from the other company would be 100(50) + 20 = $5020.
d. To solve the system of equations, we can set the two costs equal to each other: 200x + 5 = 100x + 20. By solving this equation, we find that x = 0.075. Since it doesn't make sense to buy a fraction of an item, we can round up to 1. This means the customer should buy 1 item from the other company.
I chose to solve the system of equations algebraically by setting the two costs equal to each other. This method allows me to find the specific quantity of items at which the costs are equal.
e. The solution for part d (x = 1) means that if the customer buys 1 item, the cost will be the same regardless of whether she buys from me or the other company. This point represents the break-even point for the customer, where the total cost is the same for both options.
f. The ideal number of items the customer should buy depends on factors like the customer's budget, preference, and urgency. However, based on the comparison of costs, if the customer wants to minimize the total cost, she should buy 1 item from the other company, as it will be the same cost as buying from me.
Write this like an 9th grader: Task 1:
a. In the tech business, I am willing to invest $50,000 of my own money to get started. I have decided to sell smartphones and smartwatches. The cost to make each smartphone is $200, and the cost to make each smartwatch is $100. I will sell each smartphone for $500 and each smartwatch for $250.
b. The total amount I am willing to spend on the business is $50,000. The cost per smartphone is $200, and the cost per smartwatch is $100. I can write the inequality as: 200x + 100y ≤ 50000, where x represents the number of smartphones and y represents the number of smartwatches.
c. The x-axis will represent the number of smartphones, and the y-axis will represent the number of smartwatches. The shaded region will be below and, on the line, 200x + 100y = 50000.
d. A point that falls in the shaded region could be (100, 250). The x-coordinate represents the number of smartphones (100), and the y-coordinate represents the number of smartwatches (250). The significance is that this point falls within the range of the total investment cost.
e. A point that falls directly on the line could be (250, 0). The x-coordinate represents the number of smartphones (250), and the y-coordinate represents the number of smartwatches (0). The significance is that at this point, the total investment cost is exactly the amount I am willing to spend.
f. A point that does not fall in the shaded region could be (80, 400). The x-coordinate represents the number of smartphones (80), and the y-coordinate represents the number of smartwatches (400). The significance is that this point exceeds the affordable range of the total investment cost.
Task 2:
a. I am hoping to earn at least $150,000 from selling my products.
b. To determine the price, I want to sell each item for, I need to adjust the original prices. I will sell each smartphone for $600 instead of $500, and each smartwatch for $300 instead of $250.
c. The inequality representing the desire to earn at least three times the original investment is: 600x + 300y ≥ 3(50000) = 150000, where x represents the number of smartphones and y represents the number of smartwatches.
d. The graph of the inequality will have x and y on the axes. The x-axis will represent the number of smartphones, and the y-axis will represent the number of smartwatches. The shaded region will be above and, on the line, 600x + 300y = 150000. The shaded area makes sense in this context as it represents the region where the total earnings are at least three times the original investment.
e. A point that falls in the shaded region for both inequalities could be (150, 300). The x-coordinate represents the number of smartphones (150), and the y-coordinate represents the number of smartwatches (300). The significance is that at this point, the total earnings exceed three times the original investment.
f. The ideal number of items to produce and sell depends on various factors such as market demand, production capacity, and competition. However, based on the given information, producing and selling around 150 smartphones and 300 smartwatches would be ideal as it falls within the shaded region and helps achieve the goal of earning at least three times the original investment.
Task 3:
a. Let's assume the customer wants to buy x number of items. The total cost for buying from me would be 200x + 5, where x represents the number of items bought. The total cost for buying from the other company would be 100x + 20, where x represents the number of items bought.
b. If the customer wants to buy 5 items, she should buy from the other company. The cost of buying from the other company would be 100(5) + 20 = $520, while the cost of buying from me would be 200(5) + 5 = $1005.
c. If the customer wants to buy 50 items, she should buy from me. The cost of buying from me would be 200(50) + 5 = $10,005, while the cost of buying from the other company would be 100(50) + 20 = $5020.
d. To solve the system of equations, we can set the two costs equal to each other: 200x + 5 = 100x + 20. By solving this equation, we find that x = 0.075. Since it doesn't make sense to buy a fraction of an item, we can round up to 1. This means the customer should buy 1 item from the other company.
I chose to solve the system of equations algebraically by setting the two costs equal to each other. This method allows me to find the specific quantity of items at which the costs are equal.
e. The solution for part d (x = 1) means that if the customer buys 1 item, the cost will be the same regardless of whether she buys from me or the other company. This point represents the break-even point for the customer, where the total cost is the same for both options.
f. The ideal number of items the customer should buy depends on factors like the customer's budget, preference, and urgency. However, based on the comparison of costs, if the customer wants to minimize the total cost, she should buy 1 item from the other company, as it will be the same cost as buying from me.