Given:
A manufacturer is producing two types of units Q and R.
Each unit Q costs $9 for parts and $15 for labor
Each unit R costs $6 for parts and $20 for labor
The manufacturer's budget is $810 for parts and $1800 for labor
Note: keep your eyes on the highlights numbers and the relation between the cost of the units and the allowable budget
So, we have the following equations:
[tex]\begin{gathered} 9Q+6R=810 \\ 15Q+20R=1800 \end{gathered}[/tex]We will graph the equations to find the allowable number of units:
As shown, the shaded area represents the solution to the given system
And to maximize the income we will write the income equation and test the points of the borders
Given: the income per unit is $150 for Q and $175 for R
so, the equation of the income will be:
[tex]I=150Q+175R[/tex]As shown in the figure, there are 3 points: (0, 90), (60, 45), and (90, 0)
We will test each point to find which one will give the maximum income:
[tex]\begin{gathered} Q=0,R=90\rightarrow I=150\cdot0+175\cdot90=15,750 \\ Q=60,R=45\rightarrow I=150\cdot60+175\cdot45=16,875 \\ Q=90,R=0\rightarrow I=150\cdot90+175\cdot0=13,500 \end{gathered}[/tex]So, by comparing the results:
The maximum income will be when the number of units will be as follow:
Number of units of Q = 60 unit
Number of units of R = 45 unit
The maximum income (I) = $16,875
