Otto has some whipping cream that is

Question

\[30\%\] butterfat and some milk that is
\[2\%\] 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 0.30 times x plus 0.02 times y equals 0.12 times 500. Line B is labeled x plus y equals 500. Point J is on line A and above line B.
\[\small{200}\]
\[\small{400}\]
\[\small{600}\]
\[\small{200}\]
\[\small{400}\]
\[\small{600}\]
\[\small{800}\]
\[\small{1000}\]
\[y\,\text{(milk)}\]
\[x\,\text{(cream)}\]
\[\text{line }b: x+y=500\]
\[\text{line }a: 0.30x+0.02y=0.12(500)\]
\[J\]
What does point
\[J\] represent in this context?
Choose 1 answer:
Choose 1 answer:
(Choice A) The mixture has more than the intended volume and has the intended percent of butterfat.
A
The mixture has more than the intended volume and has the intended percent of butterfat.
(Choice B) The mixture has the intended volume and has the intended percent of butterfat.
B
The mixture has the intended volume and has the intended percent of butterfat.
(Choice C) The mixture has less than the intended volume and has more than the intended percent of butterfat.
C
The mixture has less than the intended volume and has more than the intended percent of butterfat.
(Choice D) The mixture has the intended volume and has less than the intended percent of butterfat.
D
The mixture has the intended volume and has less than the intended percent of butterfat.

Answer 1

To analyze what point \( J \) represents in the context of Otto's mixture of whipping cream and milk, let's review the information provided by the equations of lines \( A \) and \( B \):

1. Equation for Line A: \( 0.30x + 0.02y = 0.12(500) \) represents the butterfat content in the mixture. When simplified, this equation indicates that Otto wants the mixture to contain \( 12% \) butterfat.

2. Equation for Line B: \( x + y = 500 \) represents the total volume of the mixture, with \( x \) being the volume of whipping cream and \( y \) being the volume of milk.

Now, let's break down what point \( J \) represents based on the settings of these equations:

• Point \( J \) is the intersection of the two lines (i.e., it satisfies both equations).
• If \( J \) lies on line \( A \) and correctly produces the butterfat percentage of \( 12% \), then from line \( B \), it will also satisfy \( x + y = 500 \).

Since point \( J \) is located where both conditions are true, it means that:

• The total volume of the mixture is \( 500, \text{mL} \) (from line \( B \)).
• This volume mixes the cream and milk in a way that results in \( 12% \) butterfat, meeting the requirements outlined in line \( A \).

Given this understanding, the correct choice among the given options is:

(Choice B) The mixture has the intended volume and has the intended percent of butterfat.