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.

Question

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.

Answer 1

a. For this business, I am willing to invest $10,000 of my own money to get started. I will sell two products - handmade candles and handcrafted soaps. The cost to make each candle is $5 and I will sell each candle for $15. The cost to make each soap is $3 and I will sell each soap for $8.

b. The total amount I am willing to spend on the business is $10,000. The cost to make each candle is $5, and the cost to make each soap is $3. Therefore, the inequality representing the fact that I cannot exceed my spending limit is 5c + 3s ≤ 10,000, where c represents the number of candles and s represents the number of soaps.

c. To graph the inequality 5c + 3s ≤ 10,000, we need to rewrite it in slope-intercept form. By isolating s, we have s ≤ -5/3c + 10/3. The graph is a straight line with a negative slope of -5/3 and a y-intercept of 10/3. We shade the region below the line.

d. Let's choose the point (400, 600) which falls in the shaded region. The x-coordinate represents the number of candles (400) and the y-coordinate represents the number of soaps (600). This point falling in the shaded region signifies that the cost of making 400 candles and 600 soaps is less than or equal to $10,000.

e. Let's choose the point (600, 700) which falls directly on the line. The x-coordinate represents the number of candles (600) and the y-coordinate represents the number of soaps (700). This point falling on the line signifies that the cost of making 600 candles and 700 soaps is exactly $10,000.

f. Let's choose the point (300, 800) which does not fall in the shaded region. The x-coordinate represents the number of candles (300) and the y-coordinate represents the number of soaps (800). This point falling outside of the shaded region signifies that the cost of making 300 candles and 800 soaps exceeds $10,000.

Task 2:
a. I am hoping to earn at least three times the amount that I originally spent on the business, which is $10,000. Therefore, I am hoping to earn at least $30,000.

b. To determine the price I want to sell each item for, I need to adjust the original prices from Task 1. After considering the desired total earnings and the cost to make each item, I decide to sell each candle for $20 and each soap for $10.

c. The inequality representing the fact that I want my total earned to be at least three times the amount originally spent is 20c + 10s ≥ 30,000.

d. To graph the inequality 20c + 10s ≥ 30,000, we need to rewrite it in slope-intercept form. By isolating s, we have s ≥ -2c + 3,000. The graph is a straight line with a negative slope of -2 and a y-intercept of 3,000. We shade the region above the line.

e. Let's choose the point (400, 600) which falls in the shaded region for both inequalities. The x-coordinate represents the number of candles (400) and the y-coordinate represents the number of soaps (600). This point falling in the shaded regions signifies that the cost of making 400 candles and 600 soaps is less than or equal to $10,000, and the total earned is at least $30,000.

f. The ideal number of items to produce and sell would be 400 candles and 600 soaps. This is because it falls within the feasible region of both inequalities, ensuring that the cost does not exceed the investment and the total earned meets the desired amount.

Task 3:
a. Let's say the customer wants to buy x items. The total cost for buying from my business would be 15x + 5, and the total cost for buying from the other company would be (0.5)(15x) + 20.

b. If the customer wants to buy 5 items, she should buy from my business. The total cost of buying from my business would be 15(5) + 5 = $80, while the total cost of buying from the other company would be (0.5)(15(5)) + 20 = $47.5 + 20 = $67.5. My business offers a better deal in this case.

c. If the customer wants to buy 50 items, she should still buy from my business. The total cost of buying from my business would be 15(50) + 5 = $755, while the total cost of buying from the other company would be (0.5)(15(50)) + 20 = $375 + 20 = $395. My business still offers a better deal in this case.

d. To solve the system of equations, we can set the two total costs equal to each other: 15x + 5 = (0.5)(15x) + 20. Simplifying the equation gives us 15x + 5 = 7.5x + 20. Subtracting 7.5x from both sides gives us 7.5x + 5 = 20. Subtracting 5 from both sides gives us 7.5x = 15. Dividing both sides by 7.5 gives us x = 2. Therefore, the customer should buy from my business if she wants to buy 2 or more items.

e. The solution x = 2 means that if the customer wants to buy 2 items, she can choose to buy from either my business or the other company, as the total cost would be the same in both cases. This point signifies the point of indifference between the two sellers.

f. The ideal number of items the customer should buy is 2. This is because at this quantity, the total cost from both my business and the other company would be the same. Any quantity larger than 2 would make my business a better choice in terms of total cost.