# Homework Help: Maximization- differential equations confused

1. Nov 6, 2005

### LakeMountD

Can someone give me a lil input on this problem.

The production rate of good chips per hour in a microelectronics lithographic production line is given by the product of throughput, V, and yield of good chips, n.

V(chips/h) = 125 - 50t + 5t^2

n= 1 / [(1+D*a)^4]

where the defect density, D, increases as the thickness, t (micrometers), of the chihp decreases: D=0.75t^-3. The active area per chip site a = 0.25cm^2. Find the thickness t that maximizes the production rate. Calculate also the optimal throughput.

[Starting with a search region of .5(< or =) t (< or =) 2.5 use the bisection method to reduce the search region to 0.5 micrometers and then switch to the Newton-Raphson method. You should use an Excel, Maple, or MATLAB program to do the calculation.

2. Nov 6, 2005

### Fermat

What is it that you are stuck on ?

3. Nov 6, 2005

### LakeMountD

I dont mean to sound like I just want an answer. I honestly just don't know where to start. Our professor basically throws this at us before teaching it so we get really confused on these word problems. Can you give me a hint on the first couple steps then i can reply with what i get stuck on? Thanks in advance.

4. Nov 6, 2005

### Fermat

Have you done maximisation problems ?

You are supposed to have a function, then differenatiate it, then set that to zero and solve. You know about that yes ?

Or, if you do know about that, is it how to tackle this particular problem ?

5. Nov 8, 2005

### LakeMountD

Yeah the problem is I don't know which one of those equations to maximize. That is the problem with I have with these word questions is that I never know which function to use and when.

6. Nov 8, 2005

### Fermat

Ok.

What you have is a maximisation problem. So you have to search the question to find out what it is that should be maximised.

"Find the thickness t that maximizes the production rate"

Voila!

So now we have to find an expression for the production rate.

Searching through the text of the question again,

"The production rate ... is given by the product of throughput, V, and yield of good chips, n."

In other words,

R = kVn

Where R is the production rate and k is a positive constant.

Since we have to find the thickness t that maximises things, then we should get V and n in terms of t.

I'm sure that you can do that bit

You should now have R as a function of t, R = R(t), which you can maximise.

Just a little comment. You will end up with a quintic in t, at^5 + bt^4 + ... = 0, which will have 5 roots. Normally you would have to investigate every single root yourself, in order to find out which one corresponded to a maximum. But the question-setter has been kind to you by telling you which interval to search in, 0.5 to 2.5.
Actually,of the 5 roots, two are complex. Of the reamining three, one is negative, so is meaningless, and of the remaining two roots I imagine one is for maximisation and the other is for minimisation.

When doing the Newton-Raphson method, unless you are quite familiar with setting up iterations in Excel, Maple, or MATLAB, then just do it by hand. It only takes a few iterations.