Math Is Fun Forum

Reuel

Member

Registered: 2010-11-28

Posts: 178

Production Units

One last problem, at least for a little while.

I happened upon this problem that was a little different from the rest. There are two given values but neither at time zero. How is a problem generally approached when there is no initial value? Or, if you assume the initial value at t=0 is 0, what data does one choose to use in order to solve for k?

"Management at a local factory recently found that the maximum number of units of some product a worker can produce in a day is 40. The rate of increase in the number of units N produced with respect to time in days by a new employee is proportional to (40 - N). Find the number of days for an employee to reach 35 units per day when on their first day he/she produced 15 units and on the 5th day he/she produced 20 units."

I set up the equation this way:

Whose general solution is

My question regards solving for C and k when the two given initial values are t=1 and t=5. Do we assume t=0 is 0 units and, if so, which of the two other data values are used to determine k? Or, if t=0 is not used, how are C and k solved for?

I am stuck and it is probably because I am tired, but I can't let go.

Thanks.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Production Units

Hi Reuel;

Actually you are just curve fitting when you are determining c and k. Not all functions are defined when the independent variable is 0. Because they never teach discrete DE's called difference equations, it is easy to never see an initial condition start at anything other than a 0. DE's are too heavily laced with physics where t = 0 is the start of every process. With difference equations you hardly ever start at 0.

Use t(1) = 15 for c just as you have done before. Then t(5) = 20 to determine k.

If you are tired than get some sleep. I will see you today!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Production Units

Thanks... I solved it that way but when I graph my solution I do not get 20 for t=5. I do, however, get 15 for t=1.

The C value I got is

and my k value was

Is that what you get?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Production Units

Hi;

I am getting:

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Production Units

hi Reuel and bobbym

I get

This is consistent with

Management at a local factory recently found that the maximum number of units of some product a worker can produce in a day is 40

I'll work on evaluating C and k.

Back soon.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Production Units

Hi guys;

Not from the DE he has got.

What I am saying is that is not the general solution of that DE. Or I am making a mistake.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Production Units

hi

Plug in t = 1, N = 15  an d also  t = 5, N = 20 to give

and

Take the C term across to the LHS and the numbers to RHS

Divide one by the other to eliminate C

So

Then you can get C from either and hence finish the question.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Production Units

hi

I get k = 0.055789
C = 26.43428
days = 29.85

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Production Units

Hi;

This is what I got.

Solve this to get k:

Substitute in and clean up.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Production Units

hi

Exactly what Reuel has in his first post.

I've just seen your edit.  Our answers are the same.  You have C = -26.43428  I think.  I had C positive.

Bob

Last edited by Bob (2011-02-03 01:55:47)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Production Units

It gets absorbed into the constant and it ends up the same. I did not realize that when I solved it at first.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Production Units

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Production Units

This conversation was amusing to come home to.

Thanks for your help guys.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Production Units

Hi Reuel;

I guess he was talking about the two of you. Anyboby mistaking me for a great mind has not been here long enough or been paying attention for the last two years. But I am a great snacker!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.