Chapter 2.6, Problem 28E

a.

To determine

To find: the representation of 3 and 2 in $f\left(x,y\right)$ .

Answer to Problem 28E

3 and 2 in the given function represents cost of salad, i.e. 3 dollars for each garlic salad and 2 dollars for each raspberry salad .

Explanation of Solution

Given information:

Oil required for one batch of garlic dressing = 2 quarts

Oil required for one batch of raspberry dressing = 3 quarts

Vinegar required for one batch of garlic dressing = 2 quarts

Vinegar required for one batch of raspberry dressing = 1 quarts

Total oil for a week = 18 quart

Total vinegar for a week = 10 quart

The function for profit is as follows:

  $f\left(x,y\right)=3x+2y$

Here x and y represents the number of batches of garlic and raspberry dressing sold respectively.

Calculation:

Because $f\left(x,y\right)$ represents profit, $3x+2y$ stand for the cost of each salad. 3 dollars is for each garlic salad ( x ) and 2 dollars for each raspberry salad ( y ).

b.

To determine

To find: the number of batches of each type of dressing should be make to maximize the profit on sales of the dressing.

Answer to Problem 28E

17

Explanation of Solution

Given information:

Oil required for one batch of garlic dressing = 2 quarts

Oil required for one batch of raspberry dressing = 3 quarts

Vinegar required for one batch of garlic dressing = 2 quarts

Vinegar required for one batch of raspberry dressing = 1 quarts

Total oil for a week = 18 quart

Total vinegar for a week = 10 quart

The function for profit is as follows:

  $f\left(x,y\right)=3x+2y$

Here x and y represents the number of batches of garlic and raspberry dressing sold respectively.

Calculation:

The inequalities derived from the given data are as follows:

  $\begin{array}{l}2x+3y\le 18\\ 2x+y\le 10\end{array}$

The graph for the inequalities is shown below.

  

The points satisfying both the inequalities are $\left(0,0\right),\left(0,6\right),\left(3,4\right)$ and $\left(5,0\right)$ . Put the points in the function $f\left(x,y\right)$ to check the maximum profit on selling salad.

  $\begin{array}{c}f\left(0,0\right)=3\left(0\right)+2\left(0\right)\\ =0\\ \\ f\left(0,6\right)=3\left(0\right)+2\left(6\right)\\ =0+12\\ =12\\ \\ f\left(3,4\right)=3\left(3\right)+2\left(4\right)\\ =9+8\\ =17\\ \\ f\left(5,0\right)=3\left(5\right)+2\left(0\right)\\ =15+0\\ =15\end{array}$

Thus the number of batches of each type of dressing should be 17 to have maximum profit