Task 1
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 to get started? How much will each item cost you to make? How much will you charge for each item?
a. 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.).
a What are we going to sell?
b What does each item cost to make or purchase?
c How much will we sell each item for?
d How much will you invest in your business?
b. 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.
a What is the inequality for your business plan?
c. Graph your inequality. Be sure to label your graph and shade the appropriate side of the line.
d. 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.
e. 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.
f. 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.
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.
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?
a Why?
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.
Equation for your company:
Equation for 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.
b What method did you choose to solve the system?
e. Explain what your solution for part D means in terms of the situation.
Task 1:
a. I am starting a business selling handmade jewelry and candles. I am willing to invest $200 of my own money to get started. It costs me $5 to make each piece of jewelry and $3 to make each candle. I plan to sell each piece of jewelry for $15 and each candle for $10.
b. Let x represent the number of jewelry pieces and y represent the number of candles made.
$5x + $3y ≤ $200
c. The graph for this inequality will show the feasible region where the cost of making jewelry and candles does not exceed $200.
d. A point that falls in the shaded region, such as (20, 40), represents making 20 pieces of jewelry and 40 candles, keeping the cost within the budget.
e. A point that falls directly on the line, such as (0, 66.67), would mean spending exactly $200 on making either 40 jewelry pieces or 66.67 candles.
f. A point that does not fall in the shaded region, such as (30, 40), would exceed the budget of $200 by spending $250 on making 30 jewelry pieces and 40 candles.
Task 2:
a. I hope to earn at least $600 from selling my products.
b. I will adjust the prices to sell each jewelry piece for $20 and each candle for $15.
$20x + $15y ≥ $600
c. The graph for this new inequality will show the feasible region where the total earned is at least three times the amount invested.
d. A point that falls in the shaded region, such as (20, 20), would mean selling 20 jewelry pieces and 20 candles to earn at least $600.
e. Choosing the same point as in Task 1 where (20, 40), with this new inequality shows the significance of both cost and earnings being met.
f. The ideal number of items to produce and sell would be 20 jewelry pieces and 20 candles, as it meets the minimum earnings requirement without exceeding the budget.
Task 3:
a.
Equation for my company: Total cost = $5x + $200 (for 5 items)
Equation for the other company: Total cost = $10x + $20 (for 5 items)
b. The customer should buy from my company as the total cost would be $225 ($5x + $200) compared to $70 ($10x + $20) from the other company.
c. If the customer wants to buy 50 items, they should still buy from my company as the total cost would be $250 ($5x + $200) compared to $520 ($10x + $20) from the other company.
d. By solving the system of equations, the customer should buy 10 items from my company to pay the same price as the other company but receive better quality products.
e. This solution means that buying more items from my company will result in a better deal for the customer in terms of cost and quality.