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.
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?
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. To earn at least three times the amount spent, we want to earn 3 times $10,000, which is $30,000.
b. To determine the new selling price for each item, we can divide the desired total earnings by the total number of items. Let's assume we will produce and sell x t-shirts and y pairs of jeans. The total number of items is x + y. So the new selling price for each item is $30,000 / (x + y).
c. The inequality representing the fact that we want our total earnings to be at least three times the amount originally spent is: (30,000 / (x + y)) * (x + y) ≥ 10,000
d. Graph of the inequality:
e. Let's choose the point (400, 200) which falls in the shaded region of both inequalities. 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. In terms of earnings, the total earnings from selling them should be at least three times the amount spent, which is $30,000.
f. The ideal number of items to produce and sell would be the one that maximizes our total earnings while still staying within the limit of our investment. This can be determined by analyzing the profit margin for each item and finding the point where the profit is maximized.
g. Let's say the chosen item is t-shirts. We can calculate the profit margin per t-shirt by subtracting the cost to make one t-shirt from the selling price of one t-shirt. In this case, it is $15 - $5 = $10 per t-shirt.
Analyze profit margin for t-shirts:
- If we produce and sell 500 t-shirts, the total profit from t-shirts would be 500 * $10 = $5,000.
- If we produce and sell 600 t-shirts, the total profit from t-shirts would be 600 * $10 = $6,000.
- If we produce and sell 700 t-shirts, the total profit from t-shirts would be 700 * $10 = $7,000.
By choosing the number of t-shirts that maximizes profit, we can determine the ideal number of items to produce and sell. In this case, producing and selling 700 t-shirts will result in the highest profit.
However, we also need to consider the profit margin for jeans and determine the ideal number of jeans to produce and sell to maximize profit. Once we have the profit margins for both t-shirts and jeans, we can compare them and make a decision on the ideal number of items to produce and sell.