Chapter 11.2, Problem 81AYU

Production To manufacture an automobile requires painting, drying, and polishing. Epsilon Motor Company produces three types of cars: the Delta, the Beta, and the Sigma. Each Delta requires 10 hours (hr) for painting, 3 hr for drying, and 2 hr for polishing. A Beta requires 16 hr for painting, 5 hr for drying, and 3 hr for polishing, and a Sigma requires 8 hr for painting, 2 hr for drying, and 1 hr for polishing. If the company has 240 hr for painting, 69 hr for drying, and 41 hr for polishing per month, how many of each type of car are produced?

To determine

To find: To manufacture an automobile requires painting, drying, and polishing. Epsilon Motor Company produces three types of cars: the Delta, the Beta, and the Sigma. Each Delta requires 10 hours (hr) for painting, 3 hr for drying, and 2 hr for polishing. A Beta requires 16 hr for painting, 5 hr for drying, and 3 hr for polishing, and a Sigma requires 8 hr for painting, 2 hr for drying, and 1 hr for polishing. If the company has 240 hr for painting, 69 hr for drying, and 41 hr for polishing per month, how many of each type of car are produced?

Answer to Problem 81AYU

Solution:

Delta type is 8 in numbers, Beta is 5 in numbers and Sigma is 10 in numbers are produced.

Explanation of Solution

Given:

To manufacture an automobile requires painting, drying, and polishing. Epsilon Motor Company produces three types of cars: the Delta, the Beta, and the Sigma. Each Delta requires 10 hours (hr) for painting, 3 hr for drying, and 2 hr for polishing. A Beta requires 16 hr for painting, 5 hr for drying, and 3 hr for polishing, and a Sigma requires 8 hr for painting, 2 hr for drying, and 1 hr for polishing. The company has 240 hr for painting, 69 hr for drying, and 41 hr for polishing per month.

Calculation:

Let the number of car produced of Delta type be $x$, Beta be $y$ and Sigma be $z$.

We have $10x + 16y + 8z = 240$

$\begin{array}{l}3x + 5y + 2z = 69\\ 2x + 3y + z = 41\end{array}$

We have $$ \left[\begin{array}{ccc}10& 16& 8\\ 3& 5& 2\\ 2& 3& 1\end{array}|\begin{array}{c}240\\ 69\\ 41\end{array}\right]$$

$\begin{array}{l}{R}_1 = \frac{R_1}{2}\\ \left[\begin{array}{ccc}5& 8& 4\\ 3& 5& 2\\ 2& 3& 1\end{array}|\begin{array}{c}120\\ 69\\ 41\end{array}\right]\end{array}$

$\begin{array}{l}{R}_2 = R_2 - R_3\\ \left[\begin{array}{ccc}5& 8& 4\\ 1& 2& 1\\ 2& 3& 1\end{array}|\begin{array}{c}120\\ 28\\ 41\end{array}\right]\\ R_2 = R_2 - R_3\end{array}$

$\begin{array}{l}\left[\begin{array}{ccc}5& 8& 4\\ -1& -1& 0\\ 2& 3& 1\end{array}|\begin{array}{c}120\\ -13\\ 41\end{array}\right]\\ R_3 = R_2 + R_3\\ \left[\begin{array}{ccc}5& 8& 4\\ -1& -1& 0\\ 1& 2& 1\end{array}|\begin{array}{c}120\\ -13\\ 28\end{array}\right]\\ R_3 = R_2 + R_3\\ \left[\begin{array}{ccc}5& 8& 4\\ -1& -1& 0\\ 0& 1& 1\end{array}|\begin{array}{c}120\\ -13\\ 15\end{array}\right]\\ R_1 = 5R_2 + R_1\\ \left[\begin{array}{ccc}0& 3& 4\\ -1& -1& 0\\ 0& 1& 1\end{array}|\begin{array}{c}55\\ -13\\ 15\end{array}\right]\\ R_3 = 4R_3 - R_1\\ \left[\begin{array}{ccc}0& 3& 4\\ -1& -1& 0\\ 0& 1& 0\end{array}|\begin{array}{c}55\\ -13\\ 5\end{array}\right]\end{array}$

We have

$\begin{array}{l}y = 5\\ - x - y = - 13\\ x = 8\\ 3y + 4z = 55\\ z = 10\end{array}$

Therefore, Delta type is 8 in numbers, Beta is 5 in numbers and Sigma is 10 in numbers are produced.