Chapter 7.4, Problem 38E

Solution

To calculate: The number of units of each food substance should she purchase in order to minimize cost and the minimum cost.

  $1.6$ units of food substance A, $9.6$ units of food substance B, $\$10.88$ .

Given Information:

Mary's diet is to contain at least 24 units of carbohydrates and 16 units of protein. Food substance A costs $1.40 per unit and each unit contains 3 units of carbohydrates and 4 units of protein. Food substance B costs $0.90 per unit and each unit contains 2 units of carbohydrates and 1 unit of protein.

Calculation:

Consider the given information,

Write the constraints as a system of inequalities using the given information.

  $\left\{\begin{array}{c}3x+2y\ge 24\\ 4x+y\ge 16\\ x\ge 0\\ y\ge 0\end{array}\right.$

Write the objective function by using the given information.

  $f=1.4x+0.9y$

Use the graphing calculator to draw the inequality and find the corner points.

  

The corner points are $\left(0,16\right),\left(1.6,9.6\right),\left(8,0\right)$ .

Now, find the value of the objective function on corner points.

Substitute 0 for x and 16 for y in the objective function.

  $\begin{array}{c}f=1.4\left(0\right)+0.9\left(16\right)\\ =14.4\end{array}$

Substitute $1.6$ for x and $9.6$ for y in the objective function.

  $\begin{array}{c}f=1.4\left(1.6\right)+0.9\left(9.6\right)\\ =10.88\end{array}$

Substitute 8 for x and 0 for y in the objective function.

  $\begin{array}{c}f=1.4\left(8\right)+0.9\left(0\right)\\ =11.2\end{array}$

The minimum value is $10.88$ .

As the feasible region is unbounded and has  $x\ge 0$  and  $y\ge 0$ , and conclude that there is no maximum value for the objective function.

Purchasing $1.6$ units of food substance A and $9.6$ units of food substance B produces the minimum cost, which is $\$10.88$ .

Therefore, $1.6$ units of food substance A, $9.6$ units of food substance B, $\$10.88$ .