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 wich is $20 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 x be the number of items the customer buys from you, and let y be the number of items the customer buys from the other company.
For your company:
Total Cost = (Number of items)*(Cost per item) + Shipping cost
Total Cost for your company = 20x + 5
For the other company:
Total Cost = (Number of items)*(Cost per item) + Shipping cost
Total Cost for the other company = (1/2)*(20y) + 20 = 10y + 20
b. To find out from whom the customer should buy 5 items, we can substitute x = 5 into the equations and compare the total costs.
Total Cost for your company = 20(5) + 5 = 105
Total Cost for the other company = 10(5) + 20 = 70
The customer should buy 5 items from the other company since the total cost is lower.
c. To find out from whom the customer should buy 50 items, we can substitute x = 50 into the equations and compare the total costs.
Total Cost for your company = 20(50) + 5 = 1005
Total Cost for the other company = 10(50) + 20 = 520
The customer should buy 50 items from the other company since the total cost is lower.
d. To solve the system of equations, we set the two total costs equal to each other and solve for x and y.
20x + 5 = 10y + 20
To simplify the equation, we can subtract 5 from both sides:
20x = 10y + 15
Now we can divide both sides by 10:
2x = y + 1.5
Now we have a linear equation in terms of x and y. We can choose to solve by substitution or elimination. In this case, substitution would be the easiest method.
From the equation 2x = y + 1.5, we can solve for y:
y = 2x - 1.5
Now we can substitute this expression for y into either equation to solve for x.
Let's substitute into the equation for your company's total cost:
20x + 5 = 10(2x - 1.5) + 20
Simplifying gives:
20x + 5 = 20x - 15 + 20
20x + 5 = 20x + 5
Since the x terms cancel out, we are left with 5 = 5. This means that the value of x can be any real number. So there are infinitely many solutions to this system.
e. The solution of 5 = 5 in terms of the situation means that for any number of items the customer buys from either company, the total costs will be the same. There is no definite answer to the question of which company the customer should buy from in this case. The customer can choose to buy from either company based on their preference or other factors, as the total costs will be equal.