Optimization problem

[bcddd214]

I am looking at this question like the monkey with the lightbulb right now.
Can someone point me in the right direction to get started so I can begin to ask an intelligent question?

All 50 rooms in a motel will be rented each night the management charges $65.00 or less per room. For each $10 increase in rent one room will remain vacant. If the cost of maintenance is $15 per room, how much should the management charge in rent to maximize daily profit?

the rate of change is 10 with respect to P (profit)
x=50?
y+65=dx?

My brain turns into jello from here.

 

[galactus]

This is not a related rates problem. So, do not worry about the "rate of change"

Like many students, you appear to be overthinking these problems. Relax and think about what profit is.

Let x=the number of rooms vacant.

profit is Revenue - Cost.

The Revenue is \(\displaystyle (65+10x)(50-x)\)

The Cost is \(\displaystyle 15x\)

So, Profit is \(\displaystyle P(x)=(65+10x)(50-x)-15x\)

The Profit function is what is to be maximized.

Differentiate, set to 0 and solve for x. Sub into the Profit function to find max profit.

 

[bcddd214]

taking the derivative of the exponents gives me p(x)=65(50) since all of the exponents are 1, they all just go to zero, is this correct?

 

[Deleted member 4993]

taking the derivative of the exponents gives me p(x)=65(50) since all of the exponents are 1, they all just go to zero, is this correct?

No...

\(\displaystyle P(x) = -10x^2......\)

 

[bcddd214]

taking the derivative of the exponents gives me p(x)=65(50) since all of the exponents are 1, they all just go to zero, is this correct?

No...

\(\displaystyle P(x) = -10x^2......\)

I am missing something, kindly elaborate.

 

[Deleted member 4993]

Please write the profit-function that Galactus derived for you - and expand it.

Please show us what do you get after expansion.

 

[bcddd214]

need a double check!

X=# of rooms
profit is rev – cost
R=np (units * cost)
10x = amount of increase
Cost of maintenance=15(50-x)
(65+10x)(50-x)
c=15x
p(x)=(65+10x)(50-x)-15x(50-x)
p(x)=65(50)+50(10x)-x(65)-10(x^2 )-15(50)-15(-x)
p(x)=2500+450x-10x^2
p ?(x)=450-20x
p ?=0 (set to 0)when x=22.5