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?
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.). Fill in the blanks below.
Answer: I am selling [product 1] and [product 2]. I am going to invest $ [dollar amount] into my business. The cost of [product 1] is $ [dollar amount] and the cost of [product 2] is $[dollar amount]. I will sell [product 1] for $ [dollar amount] and [product 2] for $[dollar amount].
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.
Answer: [product 1] is x and [product 2] is y.
Inequality: [cost of product 1]x+[cost of product 2]y ≤ [amount you are investing]
Graph your inequality. Be sure to label your graph and shade the appropriate side of the line.
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.
Answer: Ordered pair in the shaded region ([x-value],[y-value])
The x-value represents [your answer here]
The y-value represents [your answer here]
Since the point lies in the shaded region it means that [your answer here]
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.
Answer: Ordered pair on the line ([x-value],[y-value])
The x-value represents [your answer here]
The y-value represents [your answer here]
Since the point lies on the line it means that [your answer here]
Choose a point that does not fall in the shaded region. Explain what the x-coordinate and y-coordinate represent and the significance in terms of cost of this point falling outside of the shaded region.
Answer: Ordered pair is not in the shaded region ([x-value],[y-value])
The x-value represents [your answer here]
The y-value represents [your answer here]
Since the point does not lie in the shaded region it means that [your answer here]
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.
How much money are you hoping to earn from selling your products?
Answer: I am hoping to earn $[dollar amount] from selling my products.
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.
Answer: I will sell [product 1] for $ [dollar amount] and [product 2] for $[dollar amount].
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.
Answer: Original Investment $ [dollar amount] 3=[result]
Inequality: [price of product 1]x+[price of product 2]y [result]
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.
Answer:
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 y-coordinate represent and the significance in terms of both cost and money earned of this point falling in the shaded regions.
Answer: Ordered pair in the shaded region for both inequalities ([x-value],[y-value])
The x-value represents [your answer here]
The y-value represents [your answer here]
Since the point lies in the shaded region it means that [your answer here]
What is the ideal number of items you should produce and sell? Why?
Answer: The ideal number of items I should produce is [number], because [reason 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.
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.
Answer: You: [price of product 1]x+5=T
Other Company [price of product 12]x+20=T
If the customer in Alaska wants to buy 5 items, from whom should she buy? Explain your answer.
Answer: If the customer in Alaska wants to buy 5 items she should buy from [choose one: me/other company] because [reason why]
If the customer in Alaska wants to buy 50 items, from whom should she buy? Explain your answer.
Answer:If the customer in Alaska wants to buy 50 items she should buy from [choose one: me/other company] because [reason why]
Solve the system of equations from part A. What method did you choose to solve the system? Why?
Answer: Your Equation: [price of product 1]x+5=T
Other Company Equation: [price of product 12]x+20=T
Solution: ([x-value],[y-value])
I used [pick one: elimination, graphing, substitution] because [reason why]
Explain What your solution for part D means in terms of the situation.
Answer: My solution means [explain meaning]
I am selling [product 1] and [product 2]. I am going to invest $[dollar amount] into my business. The cost of [product 1] is $[dollar amount] and the cost of [product 2] is $[dollar amount]. I will sell [product 1] for $[dollar amount] and [product 2] for $[dollar amount].
Inequality: ([cost of product 1])x + ([cost of product 2])y ≤ [amount you are investing]
Graph:
(Assuming the cost of product 1 is $10, cost of product 2 is $8, and the investment is $500)
[Graph will vary based on the given values]
Point in the shaded region: (3, 20)
The x-coordinate represents the number of product 1.
The y-coordinate represents the number of product 2.
Since the point lies in the shaded region, it means that the cost of producing 3 product 1 and 20 product 2 falls within the total investment.
Point on the line: (10, 15)
The x-coordinate represents the number of product 1.
The y-coordinate represents the number of product 2.
Since the point lies on the line, it means that the cost of producing 10 product 1 and 15 product 2 exactly matches the total investment.
Point outside the shaded region: (6, 30)
The x-coordinate represents the number of product 1.
The y-coordinate represents the number of product 2.
Since the point does not lie in the shaded region, it means that producing 6 product 1 and 30 product 2 would exceed the total investment.
I am hoping to earn $[dollar amount] from selling my products.
Adjusted prices:
I will sell [product 1] for $[dollar amount] and [product 2] for $[dollar amount].
Inequality: ([price of product 1])x + ([price of product 2])y ≥ 3 * [amount you are investing]
Graph:
[Graph will vary based on the given values]
Point in the shaded region for both inequalities: (8, 12)
The x-coordinate represents the number of product 1.
The y-coordinate represents the number of product 2.
Since the point lies in the shaded region for both inequalities, it means that producing 8 product 1 and 12 product 2 falls within the total investment and leads to earning at least three times the original investment.
Ideal number of items to produce and sell: The ideal number of items I should produce is [number], because [reason why]. (This answer will vary based on the specific business and products being sold.)
Equations:
You: [price of product 1]x + 5 = T
Other Company: [price of product 2]x + 20 = T
If the customer in Alaska wants to buy 5 items, she should buy from the other company, because they have a lower price for the product and the shipping costs are the same.
If the customer in Alaska wants to buy 50 items, she should buy from me, because although the other company has a lower price per product, the shipping cost of $20 per order will significantly increase the total cost from the other company.
Solution: The method used to solve the system of equations will vary depending on the specific values given in Task 3.
The solution means that the customer's total cost will be [total cost] if she buys from me, and [total cost] if she buys from the other company, based on the number of items she wants to purchase in Alaska.