Chapter 3, Problem 26SGR

To determine

To calculate: Thenumber andtype of arrangements that the florist should make for maximizing the profit.

Answer to Problem 26SGR

For maximum profit the florist must make $0$ number of grand arrangement and $240$ number of small arrangements in one week.

Explanation of Solution

Giveninformation :

The florist can make grand arrangements in =18 minutes and profit earned is = $25

Simple arrangements in =10 minutes and profit earned is =$10

The florist work 40 hour per week

Calculation :

Consider to obtain maximum profit the florist must make x number of grand arrangement and y number of small arrangements in one week. Consider the florist can make grand arrangements in 18 minutes and simple arrangements in 10 minutes.

As the florist is allowed to work 40 hour per week therefore:

  $18x+10y\le 40\times 60$

  $18x+10y\le 2400$

The florist can make at least twice as many of the simple arrangements as the grand one therefore:

  $y\ge 2x$

As $x$ and $y$ is the number of arrangements therefore:

  $x,y\ge 0$

As the profit on the simple arrangements is $\text{\$}10$ and on the grand arrangements is $\text{\$}25$ therefore the function need to be maximized. So,

  $f(x,y)=10x+25y$

Subject to the conditions:

  $18x+10y\le 2400$

  $y\ge 2x$

  $x,y\ge 0$

Draw the feasible region bounded by the above inequalities as below:

  

Consider $\text{A},\text{B},\text{C}$ are the vertices of the feasible region. As $\text{A}$ is intersection of

  $18x+10y=2400$ , and $y=2x$ therefore,

  $\text{A}=(63.16,126.32)$

As $\text{B}$ is the intersection of $18x+10y=2400$ and $x=0$ therefore:

  $B=(0,240)$

And the other vertex is obviously:

  $C=(0,0)$

Compute the objective function $f(x,y)=10x+25y$ at $A,B,C$ as below:

  $\begin{array}{cc}f(63.16,126.32)& =10\times 63.16+25\times 126.32\\ & =3789.6\end{array}$

And,

  $\begin{array}{l}f(0,240)=10\times 0+25\times 240\\ =6000\end{array}$

And,

  $\begin{array}{l}f(0,240)=10\times 0+25\times 0\\ =0\end{array}$

Therefore, $f(x,y)=10x+25y$ is maximizedat $\text{B}=(0,240)$ .

Therefore, to obtain maximum profit the florist must make $0$ number of grand arrangement and $240$ number of small arrangements in one week.