Question

hello im.stuck.on this and need help tyR = 1.95x + 2.25yVariablesx: number of standard-mixture packagesy: number of deluxe-mixture packages

Answer 1

Variables

• x: number of standard-mixture packages

,

• y: number of deluxe-mixture packages

1.

a. From the graph, the coordinates of vertex 1 are (0, 0)

b. At vertex 2, the x- and y-coordinates of the lines 2x+3y = 300 and y = x are the same. Solving this system of equations:

`\begin{gathered} 2x+3y=300\text{ \lparen eq. 1\rparen} \\ y=x\text{ \lparen eq. 2\rparen} \\ \text{ Substituting equation 2 into equation 1} \\ 2x+3x=300 \\ 5x=300 \\ x=(300)/(5) \\ x=60 \\ \text{ Substituting x = 60 into equation 2:} \\ y=60 \end{gathered}`

The coordinates of vertex 2 are (60, 60)

c. At vertex 3, the x- and y-coordinates of the lines 2x+3y = 300 and 4x+y = 400 are the same. Solving this system of equations:

`\begin{gathered} 2x+3y=300\text{ \lparen eq. 1\rparen} \\ 4x+y=400\text{ \lparen eq. 2\rparen} \\ \text{ Multiplying equation 1 by 2 and then subtracting equation 2 from it:} \\ 2\left(2x+3y\right)=2\cdot300 \\ 4x+6y=600 \\ 4x+6y-\left(4x+y\right)=600-400 \\ 5y=200 \\ y=(200)/(5) \\ y=40 \\ \text{ Substituting y = 40 into the second equation and solving for x} \\ 4x+40=400 \\ 4x=400-40 \\ x=(360)/(4) \\ x=90 \end{gathered}`

The coordinates of vertex 3 are (90, 40)

d. From the graph, at vertex 4, the line 4x + y = 400 intersects the x-axis, then the value of the y-variable is zero. Substituting this value into the equation and solving for x:

`\begin{gathered} 4x+y=400 \\ \text{ Substituting y = 0} \\ 4x+0=400 \\ x=(400)/(4) \\ x=100 \end{gathered}`

The coordinates of vertex 4 are (100, 0)

2.

`R=1.95x+2.25y`

a. Substituting the point (0, 0) into the function R:

`\begin{gathered} R=1.95\left(0\right)+2.25\left(0\right) \\ R=0 \end{gathered}`

b. Substituting the point (60, 60) into the function R:

`\begin{gathered} R=1.95\left(60\right)+2.25\left(60\right) \\ R=252 \end{gathered}`

c. Substituting the point (90, 40) into the function R:

`\begin{gathered} R=1.95\left(90\right)+2.25\left(40\right) \\ R=265.5 \end{gathered}`

d. Substituting the point (100, 0) into the function R:

`\begin{gathered} R=1.95\left(100\right)+2.25\left(0\right) \\ R=195 \end{gathered}`

3. From item 2, the maximum revenue (265.5) corresponds to vertex 3 (90, 40). Then, she should sell 90 standard-mixture packages and 40 deluxe-mixture packages