# Maximizing Income

1. Aug 17, 2009

### Superstring

1. The problem statement, all variables and given/known data
A company makes two models of light fixtures, A and B, each of which must be assembled and packed. The time required to Assemble model A is 12 minutes, and model B takes 18 minutes. It takes 2 minutes to package model A and 1 minute to package model B. Each week there are an available 240 hours of assembly time and 20 hours for packaging.

If model A sells for $1.50 and model B sells for$1.70, how many of each model should be made to obtain the maximum weekly income?

2. Relevant equations

I remember doing problems like this in Algebra II last year. Unfortunately, I completely forget how to set up problems like this.

3. The attempt at a solution

I can't think of a way to solve it. I don't have my notes from last year either .

All I really need is the method to solving this type of problem.

Last edited: Aug 17, 2009
2. Aug 17, 2009

### tiny-tim

Welcome to PF!

Hi Superstring! Welcome to PF!
Basically, just say that there are a of A, and b of B, calculate the profit, and find an equation that specifies "Each week there are an available 240 hours of assembly time and 20 hours for packaging"

3. Aug 17, 2009

### Elucidus

This is an example of something called a Linear Programming Question. You are being asked to optimize (in this case maximize) income, which will become the objective function. You are given two features (assembling and packing) that will form constrainsts on the variables. Since these quantities represent physical objects then their number must be non-negative.

If you let a be the number of units of type A, and similarly for b, then we get:

Objective function: $\text{Income}=1.5a+1.7b$

Constraints:

Assembly: $12a+18b\leq 240(60)=14,400$

Packing: $a+2b \leq 20(60)=1200$

Practical: $x,y \geq 0$

(Note there are 60 minutes in an hour - watch your units.)

Graphing the constraints in the xy-plane will form a "freasible region," the corners of which are the only candidates for being optimal solutions (maximal in this case).

Graph the region.
Find the corners.
Test the corners in the objective function.
Determine the optimal solution point.