Integrals YAY!

[RoxieApoxie]

So, I'm basically seeing if what I am doing is correct, I think I have the right idea but I really have no idea.

The manager of a plant has been instructed to hire and train additional employees to manufacture a new product. She must hire a sufficient number of new employees so that within 30 days they will be producing 2500 units of product each day.

Because a new employee must learn an assigned task, production will increase with training. Suppose that research on similar projects indicates that production increases with training according to the learning curve, so that for the average employee, the rate of production per day is given by

DN/dt = be^-at


where N is the number units produced per day after t days of training. Because of experience with a very similar project, the manager expects the rate for this project to be

DN/dt =2.5e^-.05t

The manager tested her training program with 5 employees and learned that the average employee could produce 11 units per day after 5 days of training. On the basis of this information, she must decide how many employees to hire and begin to train so that a month from now they will be producing 2500 units of the product per day. She estimates that it will take her 10 days to hire employees. and thus she will have 15 days remaining to train them. She also expects a 10% attrition rate during this period

How many employees would you advise the plant manager to hire? check your advice by answering the following questions

1) Use the expected rate production and the results of the manager's test tp find the function relating N and t ---that is, N=N(t)

I'm assuming that this is the antiderivative, which would be N=-50e^-.05t+C, am I incorrect on this assumption?

2)Find the number of units the average employee can produce after 15 days of
training. How many such employees would be needed to maintain a production
rate of 2500 units per day?

Are we solving for C here? I guess I'm a little bit confused, I set up the equation so it would look like 2500 = -50e^-.05(15)+C and we go ahead and solve for C and that is 2500, so therefore their needs to be 2500 employees

3)Explain how you would revise this last result to account for the expected 10% attrition rate. How many new employees should the manager hire?

So ..Is that 2500 x .10 = 250 = 2500 -250 = 2250, therefore the manager should hire 2250 employees? Or.. no?


I appreciate the fact-checking!

 

[mmm4444bot]

Thank you so much for being specific, in your request for help. Most people here don't say diddly.

1) Use the expected [production rate] and the results of the manager's test [to] find the function relating N and t

I'm assuming that this is the antiderivative, which would be N = -50 e^[-.05t] + C

Yes, to go from the first derivative dN/dt to the function N(t) itself requires antidifferentiation, and you've done it correctly, but I don't think that they want a symbolic relationship between N(t) and t. They probably want you to report the actual value of C when stating the definition for function N. Use the results of the manager's test to solve for C:

N(5) = 11

2)Find the number of units the average employee can produce after 15 days of training.

This is asking for the value of N(15).


How many such employees would be needed to maintain a production rate of 2500 units per day?

The phrase "such employees" refers to employees with 15 days of training. Use your value of N(15) to calculate the requested number (that's a division problem).

After you understand questions 1 and 2, let us know what you think about question 3 (if you still need help).

Cheers ~ Mark

PS: Note the grouping symbols around the exponent which I added to quote (1) above. When texting exponents that are products, polynomials, et cetera, we must put grouping symbols around them; without grouping symbols, the meaning of these exponential terms changes (think Order of Operations, just as you must when using a graphing calculator).

 

[RoxieApoxie]

Okay, so we have the 5 employees putting out 11 units. This is the N(5)=11 answer for number one. Am I still using the antiderivative in this response, or do I use the N(5)=11 and plug it in somewhere? Ohhh I'm so confused now

Also, what exactly am I dividing for number 2? Am I using N(15)= -50 e^[-.05t] + C and dividing the antiderivative by 15...?

And I thought I understood it, apparently not LOL, thanks again for your help with this!

 

[mmm4444bot]

we have the 5 employees putting out 11 units

This statement is not worded correctly. The employees in the manager's test produced 11 units EACH. In other words, five such employees would produce a combined total of 55 units.

The number of employees tested by the manager is not relevant, in this exercise. The important part is knowing that, IF an employee receives 5 days of training, THEN that employee produces 11 units.



This is the N(5)=11 answer for number one.

No! The answer for question (1) is:

N(t) = -50 e^[-.05t] + 50



do I use the N(5)=11 and plug it in somewhere?

Yes!

Perhaps your confusion comes from not understanding function notation.

Let's review the symbol definitions.

N(t) is a variable that represents some number of units produced

t is a variable that represents some number of training days

The statement N(5) = 11 means "If an employee has 5 days of training, then they produce 11 units". This is what the manager discovered with their test.

In other words, the value of symbol N(t) is 11 when the value of symbol t is 5.

Substitute these values into the function definition for N(t) which you already found by antidifferentiation, and then solve for the constant C:

11 = -50 e^[-.05(5)] + C

You should get C = 49.94 (rounded to two decimal places, but it's probably okay to round it to the nearest Whole number).



Also, what exactly am I dividing for number 2?

You're dividing 2500 units by the number of units produced by an employee with 15 days' training.

Here's an example. Let's say that employees with 8 days' training produce 18.1 units each. How many such employees would be needed to churn out 4,000 units?

221 employees * 18.1 units per employee = 4,000 units

How do we get the answer 221? It's a division problem:

4000/18.1 = 221

 

[RoxieApoxie]

I definitely appreciate the explanation about understanding the function.

I still have more questions (sorry for bugging you, but this means a lot to mean that you're helping me actually understand this!)

For #2 which asked.

Find the number of units the average employee can produce after 15 days of
training. How many such employees would be needed to maintain a production
rate of 2500 units per day?

First, I have to find how many units the average employee can produce after 15 days of training. At first I assumed that I had to set up the equation to look like N=50e^(.05*15)+49.94 and solved for N, and that would give me 155.79, but that doesn't seem to make sense. The average employee makes 155.79 units per day ... OR do they mean total after the 15 days? I wish I could just solve an equation instead of trying to read a problem too, I'm so bad at this I apologize.

Then with 155.79 I'd have 2500/155.79 and that would equal 16.05, and that doesn't make sense either, I don't think...

Thanks again!

 

[mmm4444bot]

set up the equation to look like N = 50 e^(.05*15) + 49.94 and solved for N, and that would give me 155.79, but that doesn't seem to make sense.

You forgot the negative sign in front of -50.


The average employee makes [26.32] units per day ... OR do they mean total after the 15 days?

N(15) is the number of units produced per day by one employee after 15 days' training. The manager wants 2,500 units produced each day. How many employees will they need, if each employee produces 26.32 units per day?

 

[RoxieApoxie]

.....I forgot the negatives... ffs. OKAY, SO...

They will need 94.98 or 95 employees! HA! ..

Thank you so much!!!

For number 3 am I just taking 10% of 95? so 9.5, and 95-9.5 = 85.5 ... 86 employees?

 

[mmm4444bot]

I forgot the negatives

Ah, you're correct. I missed your missing of -0.05.



For number 3 am I just taking 10% of 95? ... 86 employees?

The manager wants production to be 2500 units.

86 employees * 26.32 units per employee = 2263 units

 

[RoxieApoxie]

Oh, I did it backwards..

Explain how you would revise this last result to account for the expected 10% attrition rate. How many new employees should the manager hire?

It's 95 x 1.10 = 104.5 = 105, so 105 employees are needed...

Thank you again for your help it means a great great great deal to me. Thanks for talking me through the steps, without you I'd be brain-farting in a corner.