# cal. word problem

• Dec 12th 2005, 06:10 PM
cobragrrll
cal. word problem
Jan is rasing money for the homeless and discovers each church group requires 2 hr of letter writing and 1 hr of followup calls, while each labor union needs 2 hr of letter writing and 3 hr of followup calls. She can raise \$150 from each church group and \$175 from each union. She has a max. of 12 hrs of letter writing and 12 hours of followup avail. each month. Determine the most profitable mixture of groups she should contact and the most money she can raise in a month.

She should contact _____ church groups and ______ labor unions to obtain \$______ in donations.
• Dec 13th 2005, 03:38 AM
ticbol
Quote:

Originally Posted by cobragrrll
Jan is rasing money for the homeless and discovers each church group requires 2 hr of letter writing and 1 hr of followup calls, while each labor union needs 2 hr of letter writing and 3 hr of followup calls. She can raise \$150 from each church group and \$175 from each union. She has a max. of 12 hrs of letter writing and 12 hours of followup avail. each month. Determine the most profitable mixture of groups she should contact and the most money she can raise in a month.

She should contact _____ church groups and ______ labor unions to obtain \$______ in donations.

This is Calculus? Umm, I can't recall now.
Let me solve this word problem through linear programming, or through a system of inequalities where the two variables cannot have negative values.

The decision variables here are number of church groups and number of unions.
Let us call these groups to be called as:
c = number of church groups to be contacted in one month.
u = number of labor union to be contacted in one month.

Problem constraints:
a) Per available number of hours of letter writing per month,
c*2 +u*2 <= 12
c +u <= 6 --------------(1)

b) Per available number of hours of follow up calls per month,
c*1 +u*3 <= 12
c +3u <= 12 ---------(2)

c)Non-negative constraints, since c or u cannot be negative,
c >= 0 -----------(3)
u >= 0 -------------(4)

Objective function:
It is about determining the most money she can raise in one month.
As for the other objective regarding most profit, the constraints do not include expenses, so we cannot determine any profit here. Only maximum collection or revenue is what we can solve for here.
Revenue, R = c*\$150 +u*\$175
R = 150c +175u ---------------objective function, in dollars.

Now, we use a graphical or geometric approach to find the feasible region, or solution region, and its corner points.
This feasible region is a polygon bounded by the 4 linear inequalities above. Its corner points, which are intersections of two of the inequalities, will give the maximum R.

I assume you know how to graph inequalities, and how to get the intersection points or corner points. Graph them using a cartesian/rectangular set of axes where, say, the vertical axis is for values of u, and the horizontal axis is for values of c.
Anyway, the feasible region is an irregular quadrilateral, whose four corner points in (c,u) are:
>>>intersection of inequalities (4) and (2) ----(0,4)
>>>intersection of inequalities (2) and (1) ----(3,3)
>>>intersection of inequalities (1) and (3) ----(6,0)
>>>intersection of inequalities (3) and (4) ----(0,0)

We test the objective function on these corner points. The highest R will show the optimal c and u:
R = 150c +175u
>>>at (0,4), R = 150*0 +175*4 = \$700.
>>>at (3,3), R = 150*3 +175*3 = \$975.
>>>at (6,0), R = 150*6 +175*0 = \$900.
>>>at (0,0), R = 150*0 +175*0 = 0.

The corner point (3,3) gave the highest R, therefore, for most money to raise in one month, she should contact 3 church groups and 3 labor unions. Maximum revenue or collection in donations is \$975 in one month. -------answer.