Question

17) Solve the system as matrix equations using inverses. 17) A company produces three models of MP3 players, models A, B, and C. Each model A machine requires 3.2 hours of electronics work, 2.8 hours of assembly time, and 4.4 hours of quality assurance time. Each model B machine requires 5.4 hours of electronics work, 2.4 hours of assembly time, and 3.4 hours of quality assurance time. Each model C machine requires 2.2 hours of electronics work, 5.8 hours of assembly time, and 4.8 hours of quality assurance time. There are 303 hours available each week for electronics, 393 hours for assembly, and 416 hours for quality assurance. How many of each model should be produced each week if all available time must be used? A) Model A: 28 B) Model A: 30 C) Model A: 30 D) Model A: 31 Model B: 22 Model B: 15 Model B: 20 Model B: 20 Model C: 45 Model C: 50 Model C: 45 Model C: 44

Answer 1

Model A=30, Model B: 20, Model C: 45

Explanation:

Let x be the amount of type A mp3 produced, and the amount of model B mp3's produced and z the amount of model C mp3 produced.

Since in total there are 303 hours for electronics work, then:

`3.2x+5.4y+2.2z=303`

Since in total there are 393 hours for assembly, then:

`2.8x+2.4y+5.8z=393`

Since in total there are 416 hours for quality assurance, then:

`4.4x+3.4y+4.8z=416`

Then, the linear system associated to the problem is

`3.2x+5.4y+2.2z=303\\2.8x+2.4y+5.8z=393\\4.4x+3.4y+4.8z=416`

with coefficient matrix
`A=\left[\begin{array}{ccc}3.2&5.4&2.2\\2.8&2.4&5.8\\4.4&3.4&4.8\end{array}\right]` and vector of constant terms
`b=\left[\begin{array}{ccc}303\\393\\416\end{array}\right]`

Since the determinant of A is equal to 36.704 then A is invertible.

Then for solve the system
`Ax=b`, is enough find the inverse of A and operate

`Ax=b\\A^(-1)Ax=A^(-1)b\\x=A^(-1)b`

Using Octave we obtain that

`A^(-1)=\left[\begin{array}{ccc}-0.22&-0.50 &0.71\\0.33&0.15&-0.34\\-0.028& 0.35&-0.20\end{array}\right]`

Then

`x=A^(-1)b=\left[\begin{array}{ccc}30\\20\\45\end{array}\right]`

This means that 30 mp3's of model A, 20 of model B and 45 of model C must be produced