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?
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.). Fill in the blanks below.
Answer: I am selling [product 1] and [product 2]. I am going to invest $ [dollar amount] into my business. The cost of [product 1] is $ [dollar amount] and the cost of [product 2] is $[dollar amount]. I will sell [product 1] for $ [dollar amount] and [product 2] for $[dollar amount].
I am selling handmade jewelry and custom-made clothing. I am going to invest $10,000 into my business. The cost of making each piece of jewelry is $5 and the cost of making each custom-made clothing item is $20. I will sell each piece of jewelry for $20 and each custom-made clothing item for $50.
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.
Answer: [product 1] is x and [product 2] is y.
Inequality: [cost of product 1]x+[cost of product 2]y ≤ [amount you are investing]
Let's say the cost of the first product (product 1) is $5 and the cost of the second product (product 2) is $20. The amount you are investing in the business is $10,000.
The inequality representing the fact that the cost of making each item should not exceed the amount invested can be written as:
5x + 20y ≤ 10000
Here, x represents the quantity of product 1 and y represents the quantity of product 2.
Graph your inequality. Be sure to label your graph and shade the appropriate side of the line.
To graph the inequality 5x + 20y ≤ 10000, we can start by plotting the boundary line.
To do this, we first need to find two points on the line. We can choose any values for x and y and solve for the corresponding values. Let's choose x = 0 first:
5(0) + 20y ≤ 10000
20y ≤ 10000
y ≤ 500
So, one point on the line is (0, 500).
Now, let's choose y = 0 and solve for x:
5x + 20(0) ≤ 10000
5x ≤ 10000
x ≤ 2000
Another point on the line is (2000, 0).
Plot these two points on a coordinate plane and draw a line through them:
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Now we need to determine which side of the line to shade. We want to represent the inequality 5x + 20y ≤ 10000, which means we want to shade the region below the line.
Shaded region:
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The shaded region below the line represents the solution to the inequality. It indicates all possible combinations of x and y values that satisfy the inequality.
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.
Answer: Ordered pair in the shaded region ([x-value],[y-value])
The x-value represents [your answer here]
The y-value represents [your answer here]
Since the point lies in the shaded region it means that [your answer here]
Let's choose the point (1000, 250) as an example, which falls in the shaded region.
The x-coordinate in this case represents the quantity of product 1, and the y-coordinate represents the quantity of product 2. So, in this situation, the x-value of 1000 means that there are 1000 units of product 1, and the y-value of 250 means that there are 250 units of product 2.
Since the point (1000, 250) lies in the shaded region, it means that this combination of quantities for product 1 and product 2 satisfies the inequality 5x + 20y ≤ 10000. In terms of cost, this means that the cost of making 1000 units of product 1 and 250 units of product 2 is less than or equal to $10,000, which is the amount invested in the business. So, the cost of producing this combination of products falls within the budget and is feasible for the business.
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.
Answer: Ordered pair on the line ([x-value],[y-value])
The x-value represents [your answer here]
The y-value represents [your answer here]
Since the point lies on the line it means that [your answer here]
Let's choose the point (800, 300) as an example, which falls directly on the line.
The x-coordinate in this case represents the quantity of product 1, and the y-coordinate represents the quantity of product 2. So, in this situation, the x-value of 800 means that there are 800 units of product 1, and the y-value of 300 means that there are 300 units of product 2.
Since the point (800, 300) lies directly on the line, it means that this combination of quantities for product 1 and product 2 satisfies the equality 5x + 20y = 10000. In terms of cost, this means that the cost of making 800 units of product 1 and 300 units of product 2 exactly matches the amount invested in the business, which is $10,000. So, the cost of producing this combination of products is precisely equal to the amount invested, indicating that the business is utilizing its resources efficiently and breaking even in terms of costs.
Choose a point that does not fall in the shaded region. Explain what the x-coordinate and y-coordinate represent and the significance in terms of cost of this point falling outside of the shaded region.
Answer: Ordered pair is not in the shaded region ([x-value],[y-value])
The x-value represents [your answer here]
The y-value represents [your answer here]
Since the point does not lie in the shaded region it means that [your answer here]