a. I am starting a new business where I am willing to invest $10,000. I have decided to sell handmade jewelry and natural skincare products. The cost to make each piece of jewelry is $5, and the sale price is $15. The cost to make each skincare product is $10, and the sale price is $25.
b. The total amount I am willing to spend on the business is $10,000. Let's represent the number of jewelry pieces as 'x' and the number of skincare products as 'y'. The inequality representing the cost limit would be:5x + 10y ≤ 10,000
c. To graph this inequality, we can create a coordinate plane with 'x' representing the quantity of jewelry pieces and 'y' representing the quantity of skincare products. The x-axis represents the number of jewelry pieces, and the y-axis represents the number of skincare products. The shaded region should be below or on the line 5x + 10y = 10,000.
d. Let's choose the point (500, 250) that falls in the shaded region. The x-coordinate represents the quantity of jewelry pieces (500) and the y-coordinate represents the quantity of skincare products (250). The significance of this point falling in the shaded region is that the total cost of producing 500 jewelry pieces and 250 skincare products is within the $10,000 investment limit.
e. Let's choose the point (1000, 500) that falls directly on the line. The x-coordinate represents the quantity of jewelry pieces (1000) and the y-coordinate represents the quantity of skincare products (500). The significance of this point falling on the line is that the total cost of producing 1000 jewelry pieces and 500 skincare products exactly matches the $10,000 investment limit.
f. Let's choose the point (3000, 400) that does not fall in the shaded region. The x-coordinate represents the quantity of jewelry pieces (3000) and the y-coordinate represents the quantity of skincare products (400). The significance of this point falling outside the shaded region is that the total cost of producing 3000 jewelry pieces and 400 skincare products exceeds the $10,000 investment limit.
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. You are hoping to earn at least three times the amount you originally spent, so you are hoping to earn 3 * $10,000 = $30,000 from selling your products.
b. To determine the price you want to sell each item for, you need to adjust the original prices. Let's say you want to sell jewelry for $20 each and skincare products for $30 each.
c. The inequality that represents the fact that you want your total earned to be at least three times the amount you originally spent would be: (20x + 30y) ≥ 3 * (5x + 10y).
d. To graph this inequality, you can create a coordinate plane with 'x' representing the quantity of jewelry pieces and 'y' representing the quantity of skincare products. The x-axis represents the number of jewelry pieces, and the y-axis represents the number of skincare products. The shaded region should be above or on the line (20x + 30y) = 3 * (5x + 10y).
e. Let's choose the point (100, 200) that falls in the shaded region for both inequalities. The x-coordinate represents the quantity of jewelry pieces (100) and the y-coordinate represents the quantity of skincare products (200). The significance of this point falling in the shaded region is that the total cost of producing 100 jewelry pieces and 200 skincare products is within the $10,000 investment limit, and the total earned from selling them are at least three times the original spent amount.
f. The ideal number of items you should produce and sell would depend on your production capacity and market demand. However, based on the original prices and adjusted prices, producing and selling more skincare products would result in higher profit margins as the sale price for skincare products is higher compared to the cost of making them. Therefore, producing and selling more skincare products could be a more profitable choice.