Question

Casper has some whipping cream that is \[18\%\] butterfat and some milk that is \[4\%\] butterfat. He wants to make a \[500\,\text{mL}\] mixture of them that is \[12\%\] butterfat. Here's a graph that shows a system of equations for this scenario where \[x\] is the volume of whipping cream he uses and \[y\] is the volume of milk he uses. A graph with two lines and a point. Line A is labeled 0.18 times x plus 0.04 times y equals 0.12 times 500. Line B is labeled x plus y equals 500. Point M is on line B and above line A. What does point \[M\] represent in this context?

Answer 1

Point M represents the ideal combination of whipping cream and milk volumes for Casper to achieve a 12% butterfat 500mL mixture.

Point M on the graph represents the specific combination of whipping cream (x) and milk (y)) volumes that Casper needs to achieve a 12% butterfat mixture in a 500mL total solution. The system of equations captures the constraints of the problem.

Line A corresponds to the butterfat content constraint. The equation
`\(0.18x + 0.04y = 0.12 * 500\)` ensures that the butterfat content in the mixture is
`\(12\%\)`. This equation is derived from the individual butterfat percentages of whipping cream (\(18\%\)) and milk (\(4\%\)) in the given proportions.

Line B represents the volume constraint, where
`\(x + y = 500\)` ensures that the total volume of the mixture is
`\(500\, \text{mL}\).`This reflects the requirement of making a
`\(500\, \text{mL}\)`mixture.

Point M is the solution to this system of equations, as it lies on Line B and above Line A. In the context of the problem, Point M specifies the exact amounts of whipping cream and milk (volumes \(x\) and \(y\), respectively) that Casper should combine to achieve the desired \(12\%\) butterfat content in the \(500\, \text{mL}\) mixture. It is the optimal mixing point that satisfies both the butterfat content and total volume constraints.