Task 1
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.
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.
Task 1:
a. For this business, I am willing to invest $10,000 of my own money to get started. I have decided to sell handmade jewelry and custom t-shirts. The cost to make each piece of jewelry is $5, and I will sell them for $20 each. The cost to make each custom t-shirt is $10, and I will sell them for $30 each.
b. The total amount I am willing to spend on the business is $10,000. Therefore, the inequality representing the fact that the cost to make each item cannot exceed this limit is:
5x + 10y ≤ 10,000
c. In order to graph this inequality, we can first rewrite it in slope-intercept form:
10y ≤ -5x + 10,000
y ≤ -0.5x + 1,000
We can then graph the line y = -0.5x + 1,000 and shade the region below the line to represent the solution to the inequality.
d. Let's choose the point (200, 0) which falls in the shaded region. The x-coordinate represents the number of handmade jewelry pieces (200) and the y-coordinate represents the number of custom t-shirts (0). The significance of this point falling in the shaded region is that the total cost to make 200 handmade jewelry pieces and 0 custom t-shirts is within the limit of $10,000.
e. Let's choose the point (0, 500) which falls directly on the line. The x-coordinate represents the number of handmade jewelry pieces (0) and the y-coordinate represents the number of custom t-shirts (500). The significance of this point falling directly on the line is that the total cost to make 0 handmade jewelry pieces and 500 custom t-shirts exactly equals the limit of $10,000.
f. Let's choose the point (300, 200) which does not fall in the shaded region. The x-coordinate represents the number of handmade jewelry pieces (300) and the y-coordinate represents the number of custom t-shirts (200). The significance of this point falling outside of the shaded region is that the total cost to make 300 handmade jewelry pieces and 200 custom t-shirts exceeds the limit of $10,000.
Task 2:
a. I am hoping to earn at least three times the amount I originally spent, which is $10,000. Therefore, I am hoping to earn a minimum of $30,000.
b. Based on the prices I originally came up with in Task 1, I may need to adjust them to achieve a total earned amount of at least $30,000. Let's say I increase the price of each handmade jewelry piece to $25 and each custom t-shirt to $35.
c. The inequality that represents the fact that I want my total earned amount to be at least three times the amount I originally spent is:
25x + 35y ≥ 30,000
d. We can graph this inequality by first rewriting it in slope-intercept form:
35y ≥ -25x + 30,000
y ≥ -0.714x + 857.14
We can then graph the line y = -0.714x + 857.14 and shade the region above the line to represent the solution to the inequality. In the context of the problem, the shaded area makes sense because it represents the combinations of handmade jewelry pieces and custom t-shirts that will result in a total earned amount of at least $30,000.
e. Let's choose the point (200, 150) which falls in the shaded region for both this inequality and the inequality found in Task 1. The x-coordinate represents the number of handmade jewelry pieces (200) and the y-coordinate represents the number of custom t-shirts (150). The significance of this point falling in the shaded regions is that the total cost to make 200 handmade jewelry pieces and 150 custom t-shirts is within the limit of $10,000, and the total earned amount will be at least $30,000.
f. The ideal number of items I should produce and sell would depend on maximizing the total earned amount while staying within the limit of $10,000. From the graph, it appears that producing and selling around 200 handmade jewelry pieces and 150 custom t-shirts would be ideal, as it falls in the shaded region of both inequalities and will result in a total earned amount of at least $30,000.
Task 3:
a. Let's represent the customer's total cost based on how many items she buys from each of the two sellers using the variables "x" for the number of items from my business and "y" for the number of items from the other company.
Total cost from my business: 20x + 5 (shipping cost)
Total cost from the other company: 10x + 20 (shipping cost)
b. If the customer in Alaska wants to buy 5 items, she should buy from my business. The total cost from my business would be 20(5) + 5 = $105, while the total cost from the other company would be 10(5) + 20 = $70. Therefore, buying from my business would be more cost-effective for the customer.
c. If the customer in Alaska wants to buy 50 items, she should still buy from my business. The total cost from my business would be 20(50) + 5 = $1,005, while the total cost from the other company would be 10(50) + 20 = $520. Therefore, buying from my business would still be more cost-effective for the customer.
d. To solve the system of equations, we can set the two expressions for the total cost equal to each other and solve for "x":
20x + 5 = 10x + 20
10x = 15
x = 1.5
Therefore, the customer would need to buy 1.5 items from each seller. Since we cannot buy fractional items, we can round up to 2 items from my business and 1 item from the other company.
I chose the substitution method to solve the system of equations because it seemed more straightforward in this case.
e. The solution for part D, which is x = 1.5, means that the customer should buy 2 items from my business and 1 item from the other company. This solution is reasonable because buying 2 items from my business at a higher price still results in a lower total cost for the customer due to the lower shipping cost.
Overall, the customer should buy from my business as it is more cost-effective for her, especially when buying a larger quantity.