A packaging company has been offered a contract to create gift boxes for perfume/cologne. The company needs to buy a special machine to make the boxes. The machine costs $3,000, and each box costs $2 for labor and materials. The perfume/cologne maker as agreed to buy each box for $5.
1.) The equation showing the packaging company's cost to make the boxes:
2.) The equation showing the perfume/cologne maker's cost to purchase the boxes:
3.) What is the solution to this system of equations?
4.) How many boxes does the perfume/cologne maker need to order for the packaging company to break even?
5.) How many boxes does the perfume/cologne maker need to order for the packaging company to make money?
6.) Use this space to show your work for solving this system of equations. Make sure to show all the steps necessary to solve the system of equations.
1) The equation showing the packaging company's cost to make the boxes:
Cost to make each box = $2 (labor and materials) + $3,000 (machine cost)
2) The equation showing the perfume/cologne maker's cost to purchase the boxes:
Cost to purchase each box = $5 (price per box)
3) Solution to this system of equations:
We can set the two equations equal to each other to find the solution:
$2 + $3,000 = $5
4) To find the number of boxes the perfume/cologne maker needs to order for the packaging company to break even, we need to find the number of boxes where the cost for the packaging company is equal to the revenue from the perfume/cologne maker. Let's denote the number of boxes as x:
Cost to make x boxes = Revenue from selling x boxes
2x + 3000 = 5x
Solving for x:
5x - 2x = 3000
3x = 3000
x = 1000
So, the perfume/cologne maker needs to order 1000 boxes for the packaging company to break even.
5) To find the number of boxes the perfume/cologne maker needs to order for the packaging company to make money, we need to find the number of boxes where the revenue from the perfume/cologne maker is greater than the cost for the packaging company. Let's denote the number of boxes as x:
Revenue from selling x boxes > Cost to make x boxes
5x > 2x + 3000
Solving for x:
5x - 2x > 3000
3x > 3000
x > 1000
So, the perfume/cologne maker needs to order more than 1000 boxes for the packaging company to make money.
6) To solve the system of equations:
$2 + $3,000 = $5
$3,002 = $5
This equation is not true, which means there is no solution for this system of equations.