A hospital performs 3 types of operations. The first type takes 30 minutes, and they charge $600 for it. The second type takes 1 hour, and they charge $900 for it. The third operation takes 2 hours, and they charge $1200 for it. The hospital can schedule up to 80 total operations per day. Moreover, they have 5 operating rooms, each of which can be used for 10 hours per day. Determine how many of each type of operation the hospital would perform to maximize its revenue.
Number of 30 minute operations = ?
Number of 1 hour operations = ?
Number of 2 hour operations = ?

Question

A hospital performs 3 types of operations. The first type takes 30 minutes, and they charge $600 for it. The second type takes 1 hour, and they charge $900 for it. The third operation takes 2 hours, and they charge $1200 for it. The hospital can schedule up to 80 total operations per day. Moreover, they have 5 operating rooms, each of which can be used for 10 hours per day. Determine how many of each type of operation the hospital would perform to maximize its revenue. 
Number of 30 minute operations =  ?
Number of 1 hour operations =  ?
Number of 2 hour operations = ?

anonymous · Answer

Thanks for the medal but can you please help me answer the question? I've been stuck on it for a while and the deadline is in 45 minutes...

rogue · Answer

I can try. What class if this for?

anonymous · Answer

Math, "linear algebra and probability". Thanks

rogue · Answer

Linear algebra... I'm afraid I can't help with that :( 
This is optimization with 3 varaibles, so it probably requires matrices or somethng.

anonymous · Answer

Yeah it does. It's alright, thanks.

accessdenied · Answer

lol, i accidentally hit the medal button
enjoy free medal.  :P

I'll see if I can at least try something here tho, I don't actually know much about this.  D:

rogue · Answer

Lol, guess I have to return the favor :P

x + y + z = 80
.5x + y + 2z = 50
profit = 600x + 900 y + 1200z
Hmm...

rogue · Answer

y = 80 - z - x

5x + 80 - z - x + 2z = 50
7x + z + 30 = 0
z = -30 - 7x

P(x,y,z) = 600x + 900y + 1200z
P(x,y,z) = 600x + 900(80 - z - x) + 1200z
P(x, z) = -300x +300z + 72000
P(x) = -300x +300(-30 - 7x) + 72000
P(x) = 63000 - 24000x

And that goes to nowhere :(

anonymous · Answer

Yeah I see what you're trying to do but it doesn't really work. This question has two variables only: A farmer has 130 acres of land suitable for cultivating crops A and B. He has enough seeds available to plant 80 acres of crop A and 110 acres of crop B. Each acre of crop A requires 14 hours of labor, and each acre of crop B requires 15 hours of labor. The farmer has at most 2100 hours of labor available. He expects to make a profit of 160 dollars per acre on crop A and 190 dollars per acre on crop B. 
How many acres should the farmer plant in order to maximize his profit?

anonymous · Answer

Think you can help with that? Please and thank you

accessdenied · Answer

ugh, sorry, I can't think of anything.  :(

rogue · Answer

I really can't help with this, I'm sorry :( I'm only taking bc calc at the moment, so this is way ahead. 
But now I see the applications of linear algebra & multivariable calc. to real world problems ;)

anonymous · Answer

It's alright! Thank you both anyway

accessdenied · Answer

i was trying to imagine something similar to what I did in algebra 2 with two-variables only and graphs, but this is much harder to think about.  :P