Chapter 11.2, Problem 82AYU

To determine

To find: A Florida juice company completes the preparation of its products by sterilizing, filling, and labeling bottles. Each case of orange juice requires 9 minutes (min) for sterilizing, 6 min for filling, and 1 min for labeling. Each case of grapefruit juice requires 10 min for sterilizing, 4 min for filling, and 2 min for labeling. Each case of tomato juice requires 12 min for sterilizing, 4 min for filling, and 1 min for labeling. If the company runs the sterilizing machine for 398 min, the filling machine for 164 min, and the labeling machine for 58 min, how many cases of each type of juice are prepared?

Answer to Problem 82AYU

Solution:

18 Orange juice ,8 grapefruit juice and 24 tomato juice are prepared.

Explanation of Solution

Given:

A Florida juice company completes the preparation of its products by sterilizing, filling, and labeling bottles. Each case of orange juice requires 9 minutes (min) for sterilizing, 6 min for filling, and 1 min for labeling. Each case of grapefruit juice requires 10 min for sterilizing, 4 min for filling, and 2 min for labeling. Each case of tomato juice requires 12 min for sterilizing, 4 min for filling, and 1 min for labeling. The company runs the sterilizing machine for 398 min, the filling machine for 164 min, and the labeling machine for 58 min.

Calculation:

Let the orange juice be $x$, grapefruit be $y$ and tomato juice be $z$.

We have

$9x + 10y + 12z = 398$

$6x + 4y + 4z = 164$

$x + 2y + z = 58$

$\left[\begin{array}{ccc}9& 10& 12\\ 6& 4& 4\\ 1& 2& 1\end{array}|\begin{array}{c}398\\ 164\\ 58\end{array}\right]$

$R_2 = \frac{R_2}{2}$

$\left[\begin{array}{ccc}9& 10& 12\\ 3& 2& 2\\ 1& 2& 1\end{array}|\begin{array}{c}398\\ 82\\ 58\end{array}\right]$

$R_2 = R_2 - R_3$

$\left[\begin{array}{ccc}9& 10& 12\\ 2& 0& 1\\ 1& 2& 1\end{array}|\begin{array}{c}398\\ 24\\ 58\end{array}\right]$

$R_3 = R_3 - R_2$

$\left[\begin{array}{ccc}9& 10& 12\\ 2& 0& 1\\ -1& 2& 0\end{array}|\begin{array}{c}398\\ 24\\ 34\end{array}\right]$

$R_1 = 12R_2 - R_1$

$\left[\begin{array}{ccc}15& -10& 0\\ 2& 0& 1\\ -1& 2& 0\end{array}|\begin{array}{c}-350\\ 24\\ 34\end{array}\right]$

$R_3 = 15R_3 + R_1$

$\left[\begin{array}{ccc}15& -10& 0\\ 2& 0& 1\\ 0& 20& 0\end{array}|\begin{array}{c}-350\\ 24\\ 160\end{array}\right]$

We have

$160 = 20y$

$y = 8$

$15x - 10y = -350$

$15x - 10\left(8\right) = -350$

$x = 18$

$x + 2y + z = 58$

$18 + 16 + z = 58$

$z = 24$

Therefore, 18 Orange juice ,8 grapefruit juice and 24 tomato juice are prepared.