Question

A manufacturer is producing two types of units. Each unit Q costs $9 for parts and $15for labor and each unit R costs $6 for parts and $20 for labor. The manufacturer'sbudget is $810 for parts and $1800 for labor. If the income per unit is $150 for Q and$175 for R, how many units of each should be manufactured to maximize income?

Answer 1

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:

`\begin{gathered} 9Q+6R=810 \\ 15Q+20R=1800 \end{gathered}`

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:

`I=150Q+175R`

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:

`\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}`

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