Quote:
Originally Posted by franksurvived
Calculus Optimization Help ASAP... Thanks in advance...?
The state has created new legislation about “crunch*‐time”, and as such requires us to adjust our projects for the next quarter. So for the next 13 weeks we will assume that there will be no scheduled overtime. Our adjusted budget for the quarter gives us 520 man*‐hours per employee.
If we have to schedule more hours, we will have to cut into the profits,
which we don’t want to do. So, with these new adjustments here are our projected needs, based on previous projects: Note: To convert from man*‐hours to number of employees, divide by 520 (which is the number of full
time hours in the quarter)
Problem 1
To create a console game, we will assume 10,920 man*‐hours of
development, 13,000 man*‐hours on art, 3,120 man*‐hours for
design, and 2,080 man*‐hours for production management.
The profit margin for a console game is 80% higher; that is, about $1.8
million for a console title and $1 million for a handheld title.
The work requirements for a handheld game quite a bit different:
7,280 man*‐hours in development, 2,600 man*‐hours art, 9,360 man*‐hours in design, and 2,600 man*‐hours in production management.
Our current staff consists of 238 programmers, 225 artists, 180 designers, and 57 production managers.
Figure out how many console and handheld games can be made this
quarter to maximize our profit. In addition, report what pools (development, artists, designers, and managers) have some unutilized employees, and which pools need to be expanded.
Note: If a console game makes 1.8x more profit than a handheld game, a handheld game makes 1x profit.
Problem 2
We have decided to expand and create a new PC games department. Out projection indicate a PC game makes 40% more profit; that is, about $1.4 million in profit. The work requirements are 9,360 man*‐hours for
development, 8,840 man*‐hours on artwork, 5,720 man*‐hours
for design, and 1,560 man*‐ hours for production management.
To help staff this department, we hired 44 more programmers for the
development team, 58 more artists for the art team, and 2 more
managers for the management team.
Figure out how many console, PC, and handheld games can be made
this quarter to maximize our profit. In addition, report what pools
(development, artists, designers, and managers) have some unutilized employees, and which pools need to be expanded.
Thanks I although need to see the work being laid out so I can understand I'm having such a hard time with these two problems. That even includes Optimization as a whole thanks for your input and answers.
30 minutes ago  4 days left to answer.

I'm not sure that you need to use calculus to solve these problems. They looked more like linear programming problems to me, so I used linear programming to solve problem 1.
d = development
a = art
s = design
p = production
Console (c= number of consoles)

d: 10920/520 = 21 people
a: 13000/520 = 25 people
s: 3120/520 = 6 people
p: 2080/520 = 4 people
Handheld (h = number of handhelds)

d: 7280/520 = 14 people
a: 2600/520 = 5 people
s: 9360/520 = 18 people
p: 2600/520 = 5 people
So our limits are given by:
Eq. 1: 21c + 14h <= 238
Eq. 2: 25c + 5h <= 225
Eq. 3: 6c + 18h <= 180
Eq. 4: 4c + 5h <= 57
Since each of these are linear equations I assumed c was the xaxis and h was the yaxis.
For each equation I found the x and y intercepts by setting c equal to 0 and solving for h, then setting h equal to 0 and solving for c.
c  h

Eq. 1: 0  17
Eq. 1: 34/3  0
Eq. 2: 0  45
Eq. 2: 9  0
Eq. 3: 0  10
Eq. 3: 30  0
Eq. 4: 0  11.4
Eq. 4: 14.25  0
I used the intercepts to find the slope of the line for each equation.
m1 = (17  0)/(0  34/3) = 1.5
m2 = (45  0)/(0  9) = 5
m3 = (10  0)/(0  30) = 1/3
m4 = (11.4  0)/(0  14.25) = 0.8
Rewriting the equations gives:
Eq. 1: h = 1.5c + 17
Eq. 2: h = 5c + 45
Eq. 3: h = c/3 + 10
Eq. 4: h = 0.8c + 11.4
I then graphed them to see where they intersected each other and the x and y axes. There were 4 points which are easy to see if you graph them.
Point1: (0,10)
Point2: (3,9)
Point3: (8,5)
Point4: (9,0)
Three of the lines (1, 2, and 4) intersect at (8,5).
Then plug in the values for c and h into the equation
Profit = 1.8c + 1h
And pick the largest to find the maximum profit.
Profit1 = 1.8(0) + 10 = 10 million
Profit2 = 1.8(3) + 9 = 14.4 million
Profit3 = 1.8(8) + 5 = 19.4 million
Profit4 = 1.8(9) + 0 = 16.2 million
So the max profit occurs when 8 consoles and 5 handhelds are produced