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
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 y coordinate 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. 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. Let's choose the point (600, 300) which 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 ideal number of items I should produce and sell would depend on various factors such as demand, market conditions, and production capacity. However, producing and selling more items would increase the chances of reaching or exceeding the goal of earning at least three times the amount originally spent. Therefore, I should aim to produce and sell as many items as is feasible and profitable for my clothing business.