Task A
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.
a. For my business, I am willing to invest $10,000 of my own money to get started. I have decided to sell handmade jewelry and custom-made leather bags. The cost to make each piece of jewelry is $5, and I will sell them for $25 each. The cost to make each leather bag is $50, and I will sell them for $150 each.
b. Based on the total amount I am willing to spend on the business, which is $10,000, and the costs per item, the inequality that represents the fact that while making each item I can't exceed this limit is:
5x + 50y ≤ 10,000
where x represents the number of pieces of jewelry and y represents the number of leather bags.
c. Graph of the inequality:
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3000|____________________________________________
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2000|
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1000|----(0,200)-----------------(2000,0)-----
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--------------------------------------
0 2000 4000 6000 8000 10000
d. Let's choose the point (100, 100) that falls in the shaded region. In this case, the x-coordinate represents the number of pieces of jewelry (100), and the y-coordinate represents the number of leather bags (100). The significance of this point falling in the shaded region is that the cost of producing 100 pieces of jewelry (100 * $5 = $500) and 100 leather bags (100 * $50 = $5,000) is within the budget of $10,000.
e. Let's choose the point (2000, 0) that falls directly on the line. In this case, the x-coordinate represents the number of pieces of jewelry (2000), and the y-coordinate represents the number of leather bags (0). The significance of this point falling directly on the line is that the cost of producing 2000 pieces of jewelry (2000 * $5 = $10,000) reaches the maximum limit of the investment budget.
f. Let's choose the point (300, 200) that does not fall in the shaded region. In this case, the x-coordinate represents the number of pieces of jewelry (300), and the y-coordinate represents the number of leather bags (200). The significance of this point falling outside of the shaded region is that the cost of producing 300 pieces of jewelry (300 * $5 = $1,500) and 200 leather bags (200 * $50 = $10,000) exceeds the investment budget of $10,000.
can you do task b
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?
a. To ensure that my total amount earned is at least three times the amount I originally spent (which is $10,000), I am hoping to earn at least $30,000 from selling my products.
b. In order to achieve this earnings goal, I will need to adjust the original prices of the items. I will sell each piece of jewelry for $35 (increased from $25) and each leather bag for $180 (increased from $150).
c. The inequality that represents my earning goal is:
35x + 180y ≥ 30,000
where x represents the number of pieces of jewelry and y represents the number of leather bags.
d. Graph of the inequality:
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5000|____________________________________________
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4000|
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3000|----- (500,300)-----------------------------
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2000|
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--------------------------------------
0 1000 2000 3000 4000 5000
The shaded area on the graph makes sense in the context of the problem. It represents the region where the total earnings are at least three times the original investment.
e. Let's choose the point (500, 300) that falls in the shaded region for both inequalities. In terms of cost, the x-coordinate represents the number of pieces of jewelry (500), and the y-coordinate represents the number of leather bags (300). The significance of this point falling in the shaded regions is that the cost of producing 500 pieces of jewelry (500 * $35 = $17,500) and 300 leather bags (300 * $180 = $54,000) falls within the budget of $10,000. In terms of money earned, this point represents a total of ($35 * 500) + ($180 * 300) = $26,500, which is greater than the desired goal of $30,000.
f. The ideal number of items to produce and sell depends on various factors such as market demand, production capacity, and profitability. In this case, producing and selling a combination of 500 pieces of jewelry and 300 leather bags would generate a total revenue of $26,500, which is less than the desired goal of $30,000. Therefore, it might be necessary to adjust the production quantities and prices further to reach the ideal number of items that maximize both costs and earnings.
now can you do task c
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. Let's represent the number of items the customer wants to buy as "x".
For your business:
Customer’s total cost from you = (cost per item * number of items) + shipping cost
Customer’s total cost from you = (35x) + 5
For the other company:
Customer’s total cost from the other company = (cost per item * number of items) + shipping cost
Customer’s total cost from the other company = (17.50x) + 20
b. If the customer in Alaska wants to buy 5 items, we can calculate the total cost from each seller:
Customer’s total cost from you = (35 * 5) + 5 = $180
Customer’s total cost from the other company = (17.50 * 5) + 20 = $102.50
In this case, the customer should buy from the other company because the total cost is lower.
c. If the customer in Alaska wants to buy 50 items:
Customer’s total cost from you = (35 * 50) + 5 = $1,755
Customer’s total cost from the other company = (17.50 * 50) + 20 = $895
In this case, the customer should buy from the other company because the total cost is still significantly lower.
d. To solve the system of equations:
(35x) + 5 = (17.50x) + 20
I chose to use the method of elimination to solve this system. By subtracting (17.50x) from both sides and subtracting 5 from both sides, we get:
35x - 17.50x = 20 - 5
17.50x = 15
x = 15 / 17.50
x = 0.857
e. The solution x = 0.857 means that the customer should buy approximately 1 item from your business and the remaining items from the other company. This is because the cost of shipping from your business is lower, but the cost per item from the other company is lower. So, to minimize the total cost, the customer should purchase the majority of the items from the other company and only a few items from your business.