MATRIX help: numbers of widgets to maximize profits

[KingAce]

ok so first off this is the question:

A manufacturer of metal widgets makes two types, A and B. Each type A widget requires 0.5 hours of cutting ,1.5 hours of welding, and 1 hour of finishing. Each type B widget requires 0.5 hours of cutting, 1 hour of welding, and 1.5 hours of finishing. Each day, 15 hours of cutting time, 36 hours of welding time, and 42 hours of finishing time are available. If the profits on a type A widget and on a type B widget are $3 and $4 respectively, how many of each type of widget should be made each day to maximize total profits?

My work (what I got up to):

15hrs >(or equal to) .5A +.5B
36hrs >(or equal to) (3/2)A + 1B
42hrs >(or equal to) 1A + (3/2)B
A>(or equal to) 0
B>(or equal to) 0

Profits/Day = A($3) + B($4)

What would I do next? Could I possibly plug it into a matrix on my calculator? But then how would I solve it after?

Thanks for the help!

 

[masters]

I believe I would look at this as a linear programming problem with the constraints you have laid out.

\(\displaystyle .5A+.5B\leq15\)

\(\displaystyle 1.5A+B\leq36\)

\(\displaystyle A+1.5B\leq42\)

\(\displaystyle A\geq0\)

\(\displaystyle B\geq0\)

The profit function would then be \(\displaystyle f(A,B)=3A+4B\)

Graph the system of inequalities . The intersection of the graphs is called the feasible region. The maximum value of the profit function will always occur at one of the vertices.