Task 1
A. I have decided to start a clothing business where I will sell both t-shirts and jeans. I am willing to invest $10,000 of my own money to get started. The cost per t-shirt is $5 to make, and I will charge $15 for each t-shirt. The cost per pair of jeans is $20 to make, and I will charge $50 for each pair of jeans.
B. The total amount I am willing to spend on the business is $10,000. So, the inequality representing the fact that I can't exceed this limit while making each item would be:
5x + 20y ≤ 10,000
where x represents the number of t-shirts and y represents the number of pairs of jeans.
C. Graph of the inequality:
D. Let's choose the point (400, 200) which falls in the shaded region. The x-coordinate represents 400 t-shirts, and the y-coordinate represents 200 pairs of jeans. The significance in terms of cost is that the total cost to make 400 t-shirts and 200 pairs of jeans is within the limit of $10,000.
E. Let's choose the point (400, 150) which falls directly on the line. The x-coordinate represents 400 t-shirts, and the y-coordinate represents 150 pairs of jeans. The significance in terms of cost is that the total cost to make 400 t-shirts and 150 pairs of jeans exactly matches the limit of $10,000.
F. Let's choose the point (300, 400) which does not fall in the shaded region. The x-coordinate represents 300 t-shirts, and the y-coordinate represents 400 pairs of jeans. The significance in terms of cost is that the total cost to make 300 t-shirts and 400 pairs of jeans exceeds the limit of $10,000.
Task 2
A. The amount of money I am hoping to earn from selling my products is at least three times the amount I originally spent, so it would be $10,000 x 3 = $30,000.
B. To determine the price I want to sell each item for, I will adjust the original prices mentioned in Task 1. Let's say I increase the price of each t-shirt to $20 and the price of each pair of jeans to $60.
C. The inequality representing the fact that I want my total earned to be at least three times the amount I originally spent would be: 20x + 60y ≥ 30,000, where x represents the number of t-shirts and y represents the number of pairs of jeans.
D. Graph of the new inequality:
E. the point (600, 300) falls in the shaded region for both this inequality and the inequality found in Task 1. The x-coordinate represents 600 t-shirts, and the y-coordinate represents 300 pairs of jeans. The significance in terms of cost is that the total cost to make 600 t-shirts and 300 pairs of jeans is within the limit of $10,000, and in terms of money earned, the total earned from selling these items is at least three times the amount originally spent.
F. The amount of things I should make and sell depends on how many people want them, how well the market is doing, and how much I can produce. However, if we make and sell more things, we will have a better chance of making at least three times the money we spent. So, I need to make and sell as many clothes as I can while still making a profit for my business.
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. Let x represent the number of items the customer buys from you and y represent the number of items the customer buys from the other company.
The total cost from your company would be: 5x + 5
The total cost from the other company would be: (0.5x) + 20
B. If the customer in Alaska wants to buy 5 items, she should buy from the other company. The total cost from your company would be 5(5) + 5 = $30, while the total cost from the other company would be (0.5(5)) + 20 = $22.50. The customer would save money by buying from the other company.
C. If the customer in Alaska wants to buy 50 items, she should buy from your company. The total cost from your company would be 5(50) + 5 = $255, while the total cost from the other company would be (0.5(50)) + 20 = $45. The customer would save money by buying from your company.
D. To solve the system of equations, we can set the two total costs equal to each other and solve for x:
5x + 5 = (0.5x) + 20
Simplifying, we get:
4.5x = 15
x = 15/4.5
x ≈ 3.33
To solve for y, we can substitute the value of x back into one of the equations. Let's use the equation for the total cost from the other company:
y = (0.5(3.33)) + 20
y ≈ 21.67
Therefore, the customer should buy approximately 3 items from you and approximately 22 items from the other company.
I chose to use the substitution method to solve the system of equations because it allows for easy substitution and simplification.
E. The solution for part D means that if the customer buys approximately 3 items from you and approximately 22 items from the other company, the total cost from both sellers will be equal. This is the point where it doesn't matter from whom the customer buys, as the total cost will be the same.