Deriving an equation

[jbone]

I have spent a long time trying to figure out how to get started on this question. The only help I need is part a; I can do part b and c once I figure out the equation for part a.
Here is the question:

One very important question facing hospitals is this: How big must a hospital be (in
terms of patient-days of care) to minimize the cost per patient-day? According to one
well-known study, the total cost (in dollars) of operating a hospital (of a particular
type) can be approximated by
C = 4,700,000 + 0.00013X^2
;
where X represents the number of patient-days.
(a) Derive a formula for the relationship between cost per patient-day and the number
of patient days.
(b) On the basis of the results of this study, how big must a hospital be (in terms of
patient-days) to minimize the cost per patient-day?
(c) Show that your result minimizes, rather than maximizes, the cost per patient-day.


Thanks for the help!

 

[HallsofIvy]

You are told that the cost is C = 4,700,000 + 0.00013X^2 and the number of days is X. To find the "cost per day", divide by X.

 

[jbone]

So I just divide the entire formula by X?

C=(4,700,000+.00013x^2)/x is my answer for part a?

Then just take the first derivative and set it equal to zero for part b

And take second derivative to prove it's a minimum it part c?


I am still getting a weird answer.

 

[JeffM]

So I just divide the entire formula by X?

C=(4,700,000+.00013x^2)/x is my answer for part a?

Then just take the first derivative and set it equal to zero for part b

And take second derivative to prove it's a minimum it part c?


I am still getting a weird answer.

Hard to tell whether it is a weird answer when you do not show what it is or how you got it.

 

[jbone]

My working out was incorrect.

I appreciate the help.

 

[Boko]

I am trying to brush up on my calculus, so this question comes in handy.

my answer for b) -4,700,000/x^2 + 0.00013 = 0 --> x= 190,141.648

for part c) if I take the second derivative, I get 4,700,000/x^3

What do I do now? Plug in 190,141.68 from part b) ? It comes out to be an extremely large number.

I don't really have an intuition about the numbers in this problem. Can somehow help me, please?

 

[Deleted member 4993]

I am trying to brush up on my calculus, so this question comes in handy.

my answer for b) -4,700,000/x^2 + 0.00013 = 0 --> x= 190,141.648

for part c) if I take the second derivative, I get 4,700,000/x^3

What do I do now? Plug in 190,141.68 from part b) ? It comes out to be an extremely large number.

I don't really have an intuition about the numbers in this problem. Can somehow help me, please?

So the second derivative will be always positive since x is positive.

Then at x = 190142 you have a minima.

 

[Boko]

Oh, ok. Thank you. I was thinking in a different direction.