Chapter 3, Problem 26SGR

To determine

To calculate: The number and type of arrangements produced by florist to maximize the profit.

Answer to Problem 26SGR

The florist should do 63 grand arrangements and 126 simple arrangements.

Explanation of Solution

Given information:

Time taken to make a grand arrangement is 18 minutes and a simple arrangement is 10 minutes. She makes at least twice as many of the simple arrangements as the grand arrangements. The number of working hours in a week are 40 hours.

The profit made on simple arrangement is $10 and on grand arrangement is $25.

Calculation:

Consider the provided information thattime taken to make a grand arrangement is 18 minutes and a simple arrangement is 10 minutes. She makes at least twice as many of the simple arrangements as the grand arrangements. The number of working hours in a week are 40 hours.

The profit made on simple arrangement is $10 and on grand arrangement is $25.

Let x denote the number of grand arrangements and y denote the number of simple arrangements.

Since, time taken to make a grand arrangement is 18 minutes and a simple arrangement is 10 minutes. The number of working hours in a week are 40 hours.

  $\begin{array}{c}18x+10y\le 60\times 40\\ 18x+10y\le 2400\end{array}$

She makes at least twice as many of the simple arrangements as the grand arrangements.

Therefore,

  $\begin{array}{c}y\ge 2x\\ x\ge 0\\ y\ge 0\end{array}$ .

Plot the inequalities and shade the common region $y\ge 2x,x\ge 0,y\ge 0\text{ and }18x+10y\le 2400$

  

The profit function that is to be maximized is $f\left(x,y\right)=25x+10y$ .

Substitute the vertices of the feasible region to find the point at which maximum revenue is there.

Substitute $\left(0,0\right)$ in the function $f\left(x,y\right)=25x+10y$ .

  $f\left(0,0\right)=0$

Substitute $\left(0,240\right)$ in the function $f\left(x,y\right)=25x+10y$ .

  $f\left(0,240\right)=2400$

Substitute $\left(63,126\right)$ in the function $f\left(x,y\right)=25x+10y$ .

  $f\left(63,126\right)=2835$

Since, maximum profit is $2835 which is when florist should do 63 grand arrangements and 126 simple arrangements.