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 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. The total amount I am hoping to earn from selling my products is at least three times the amount I originally spent.
b. To determine the price I want to sell each item for, I will need to adjust the original prices that I came up with in Task 1. Let's say I originally priced my items at $10 and $20. I will need to increase these prices to ensure that the total amount earned is at least three times the original spending.
c. The inequality that represents the fact that I want my total earned to be at least three times the amount that I originally spent is:
Total Earned ≥ 3 * Total Spending
This can be written as:
(Price of Item 1 * Quantity of Item 1 + Price of Item 2 * Quantity of Item 2) ≥ 3 * (Original Price of Item 1 * Quantity of Item 1 + Original Price of Item 2 * Quantity of Item 2)
d. To graph the inequality, we will need to assign a variable for the price of each item. Let's use x for the price of Item 1 and y for the price of Item 2.
The inequality becomes:
(x * Quantity of Item 1 + y * Quantity of Item 2) ≥ 3 * (Original Price of Item 1 * Quantity of Item 1 + Original Price of Item 2 * Quantity of Item 2)
To graph this inequality, we need to plot the line:
(x * Quantity of Item 1 + y * Quantity of Item 2) = 3 * (Original Price of Item 1 * Quantity of Item 1 + Original Price of Item 2 * Quantity of Item 2)
Then, we shade the region that is above or on the line, as this represents the values that satisfy the inequality. The shaded area should make sense in the context of the problem.
If the shaded area does not make sense, we may need to adjust the original values (the investment and the sale price for both products) so that the shaded area is reasonable.
e. Let's choose a point that falls in the shaded region for both this inequality and the inequality found in Task 1. Let's say we choose the point (10, 15).
In this case, the x-coordinate represents the price of Item 1, and the y-coordinate represents the price of Item 2. The significance of this point falling in the shaded region is that selling Item 1 for $10 and Item 2 for $15 allows us to earn at least three times the amount we initially spent.
f. The ideal number of items I should produce and sell would depend on the specific values of the prices and quantities. To determine the ideal number of items, I would need more information about the specific prices, quantities, and original spending amounts.
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.
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. Let's assume the price of the item from my company is $X.
The customer's total cost when buying from my company would be:
Total Cost = (Price of Item from my company * Quantity of Items) + Shipping Cost ($5)
This can be written as:
Total Cost = X * Quantity + $5
Let's assume the price of the item from the other company is $Y.
The customer's total cost when buying from the other company would be:
Total Cost = (Price of Item from the other company * Quantity of Items) + Shipping Cost ($20)
This can be written as:
Total Cost = Y * Quantity + $20
b. If the customer in Alaska wants to buy 5 items, she should buy from the other company.
To determine this, we compare the total cost from my company to the total cost from the other company.
Total Cost from my company = X * 5 + $5
Total Cost from the other company = Y * 5 + $20
If the other company charges half of my price, then Y = 0.5X.
Comparing the total costs:
X * 5 + $5 < Y * 5 + $20
5X + $5 < 5(0.5X) + $20
5X + $5 < 2.5X + $20
2.5X > $15
X > $6
Since my total cost is more expensive than the other company's for 5 items, the customer in Alaska should buy from the other company.
c. If the customer in Alaska wants to buy 50 items, she should still buy from the other company.
Following the same logic as in part b, if the other company charges half of my price, then Y = 0.5X.
Comparing the total costs:
X * 50 + $5 < Y * 50 + $20
50X + $5 < 50(0.5X) + $20
50X + $5 < 25X + $20
25X > $15
X > $0.6
Since my total cost is more expensive than the other company's for 50 items, the customer in Alaska should still buy from the other company.
d. To solve the system of equations from part a, I will use substitution.
Substituting Y = 0.5X into the second equation:
Total cost = (Price of Item from the other company * Quantity of Items) + $20
Total Cost = (0.5X * Quantity) + $20
Equating both Total Cost equations:
X * Quantity + $5 = (0.5X * Quantity) + $20
Simplifying the equation:
X * Quantity - 0.5X * Quantity = $20 - $5
0.5X * Quantity = $15
Quantity = $15 / 0.5X
Quantity = 30 / X
e. The solution from part d represents the breakeven point where the total cost from my company is equal to the total cost from the other company. In terms of the situation, it means that if the customer purchases a quantity greater than or equal to 30/X items, it would be more cost-effective to buy from my company. If the customer purchases a quantity less than 30/X items, it would be more cost-effective to buy from the other company.